Dealer is pressuring me to return the leased car and lease another car

I feel like they are pressuring me to do something I don't want to. Not to mention they are literally calling my number twice a week leaving a voicemail.

Of course they are, they want to make money by selling it themselves and more money by selling you a new lease.

What option do I have?

If your lease agreement specifies a post-lease purchase price and you like the car and/or that price then it sounds like your option is clear, get a loan (you can get a quote from the dealership but should also check out other lenders for the best rate) and buy the car.

I don't want to be rude to them...

You owe them nothing, they are lucky to have your business. They don't care if they're making you uncomfortable, they're trying to make money off you.

As for forcing you to return the car and anything about damages is concerned that just sounds like things they are saying to get you into a new lease and to sell your current car themselves. Ultimately though it's about what is in your lease terms, typically the option to purchase post-lease is yours and not subject to their approval and damages would be irrelevant if you are buying it rather than returning it.

Be direct. "I would like to purchase the car, and am happy to talk financing options with you. I do not want to return the car." Just say that, every time they call.

Then, as Hart said, do some research into financing through banks - either your bank, a local credit union/bank, or a big national bank; each has its advantages, I tend to go in that order. If your bank is one you're happy with and they'll give you a decent rate, take that. They'll give you a certain quote given the car, the age, and the money, and take that to the dealer. If the dealer can beat it, great - you can take their offer instead if you like, or you can just take your already-given rate if they can't (or if you'd rather not do business with them, of course, you can take the bank's rate anyway).

Sales is all about pressuring you to do what they want - you have to be firm and just tell them what you want and stick to it. Remove emotion from it; don't be intentionally rude per se, but don't try to be nice either. Just tell them what you want, and what you're going to do, and that you aren't going to listen to their arguments.

As for damages, it seems the opposite normally for me - if you return a car damaged you'll be liable for them, but not if you buy the car for the agreed-upon price - so that's probably just a sales tactic.

There is a shortage of used cars on the market right now as a result of the pandemic. It's an incredibly hot market for used cars, and the dealer is trying to secure what sounds like a very lightly-used car so they can make a nice profit. I've been regularly hearing of people being bought out of their leases early just so the dealers can have the additional used inventory.

You're not obligated to move into a new lease. But seeing as you know they are fairly desperate to get your vehicle, you could use that to your advantage to negotiate a new lease. I know you aren't planning on that route, but it might be worth considering if you can get the newest model for a great lease deal.

Ultimately, do what's best for you. They can't force you to return the car.

It's very likely that your lease is with a finance company (e.g. GM Financial for GM cars) and the dealer isn't really involved. Typically, at the end of a lease, you can return your car to any dealer, not just the original dealer. If your lease allows that (read it, it probably does), then you can completely ignore everything the dealer is telling you. Furthermore, if you want to buy the car, you will negotiate that through the leasing company, not the dealer.

This reframes your situation as: dealer is desperate to separate you from your money, but is completely powerless and uninvolved.

Trying to be polite to a car dealer is the surest way of losing money. Don't don't worry about it, they have the thickest skins of any profession, surpassed only by repo men and tow truck drivers, especially since he's bullshitting you about damages. They DO need to do an inspection, when they transfer ownership, but that's standard for legal reasons and it's only a couple hundred bucks.

Given your milage, a buy out would be a winning bet even before the pandemic + chip shortage and now it's a complete no brainier.

Don't bother getting financing through them, just get pre-approved for the exact payout amount in your contact through your bank or a third party lender and go to any other dealership(as other posters have pointed out you can do that, think about it: if you couldn't do it, you'd have to go back to your original dealership to turn in your lease if you moved to another state, which happens all the time) and do the buy out there. If also recommend doing it immediately and not on the last day, so there's no time pressure, you can buy out early at any time.

P.S. if you've got money in a 401k, you can take a loan against that, the rates are better and your pay the interest back to yourself.

Your choices are clear:

1. If you like the car and want to keep it, then buy it. The purchase price is in your contract from 3 years ago and they can't charge you more than that. Since you're 26K under mileage, you're very likely to be getting a bargain. (Sounds like you think it's $7K-ish.) Any dealer of your car manufacturer can assist you with the sale, or call the phone number listed in your leasing portal and ask them how to purchase.
2. If you don't want to keep the car, call 3 different dealers that lease your car type where you can return the car, and ask them how much they will pay you for you to return it to them. Note they may not be allowed to just give you the money; but they may be able to give you a substantial credit towards the purchase or lease of another car from their dealership.
3. If you don't want to keep the car, call 3 different dealers that don't lease your car type. CarMax is a good place to start. They may handle all the paperwork for buying out the lease and just hand you a check for $3-5K. (Assuming your $7K number is accurate.)

If you want option #1, don't let any dealership pressure you into buying something else. Tell them:

I would like you to assist me with buying my vehicle. If you can't help me with that, please find me someone else who can.

Repeat as necessary. (I'd be surprised if you have to say that more than twice.)

You shouldn't be getting financing from a dealer

You should be going to your local bank(s). Lots of people write financing for cars, but the dealers take more profit than anyone, so their deals are usually more expensive than anything else.

Sometimes their partnerships with the highly motivated manufacturers can yield a better bargain, but that only applies to new cars. Once you're in the used realm, banks are always a better deal. So secure the financing needed to pay off your lease.

However, avoid the "vicious cycle"

Some people choose dealers because dealers "solve every problem". For instance, if you trade in a lease that is "upside down" (you owe more on it than it's worth), dealers have strategies to "fold that debt" into the new car financing or lease... and since people are already on their back foot when starting that lease, naturally they finish that lease in the hole also.

Of course this is nice for the debtor because it means a new car every 3-4 years. But as car payments rise, it becomes an increasingly perilous financial situation for the debtor, who now does not have the cash flow margins to deal with calamity like a job loss or major repair bill. Very often there is not a possible exit to this crazy-train short of enduring a repo, credit burn, having no car, trying to keep employment without a car, etc. And often this exit is forced because of a major repair, job loss, or other financial blow.

If nobody will lend you except the dealer, that's the universe warning you

The thing about this "death spiral" refinancing is, normal banks won't do it. They will not loan you money in a way where you would owe more than the car is worth. So they serve as "your conscience": if the bank won't make the loan, you shouldn't take the loan!

The dealer pressure is a sales job: they'll make sick money if you assent.

Hey, if I could make $5000 by getting you to say "yes", I'd be blowing up your phone too.

As such, they are acting exactly as salesmen "ought to" (from their stockholders' points of view). Like anyone who calls YOU wanting to do a deal, the deal is highly advantageous to them (I wouldn't blow up your phone for $50, say) and that usually means less or not advantageous to you.

Don't be a lemming; don't ever let salesmen push you into what to do.