Can I use UPS to send my tax returns from abroad to IRS Austin?

Look at the IRS address carefully. If it is a P.O. Box, or the address is merely "Internal Revenue Service, Austin TX nine-digit zipcode" then UPS will not work because UPS and FedEx and DHL etc need a street address, and cannot deliver to a P.O. Box or to the other form of address where the nine-digit zipcode is basically a P.O. Box number; the US Postal Service is going to put every piece of mail with that nine-digit zipcode into one P.O. Box (actually, deliver to the address that the IRS has set up instead of the IRS sending out its own courier in a van to get its mail from a physical Post Office Box located in a post office). In such cases, your on-line program might tell you a street address that can be used for delivery via UPS etc, or if not, you can get the address from the IRS website.

For domestic deliveries of small packages that don't have guaranteed delivery dates, UPS seems to have a deal with the USPS that UPS will deliver the package to the local post office and then USPS will actually bring it to your home. Whether this allows a shipper to send a package via UPS to a P.O. Box or to a nine-digit zipcode that is essentially a P.O. Box is something I don't know. Whether this facility is available for international shipments via UPS is also something I don't know; the answer might by Yes in the first instance and No in the second, or No in both cases, or (most unlikely) Yes in both cases.

From the IRS website -

Austin - Internal Revenue Submission Processing Center
3651 S IH35
Austin, TX 78741

You are not the first person needing to ship to IRS via Fed-Ex or other private carrier.

For New Jersey, the shipping address is:

State of New Jersey
Division of Taxation
Revenue Processing Center
200 Woolverton Street
Building 20 
Trenton, NJ 08611

This is not a direct answer to your question, but I am suggesting an alternative. FreeFillableForms.com is endorsed by the IRS (https://www.irs.gov/e-file-providers/before-starting-free-file-fillable-forms) and allows you to e-file, which is the arguably fastest, safest way to get your return to the IRS (I am surprised your online service did not give you that option). They have most forms available, but in case you need exotic ones, double check before your start. Be aware that the IRS does some sanity checks on e-filed returns before accepting them, so you might get an e-mail a day after filing that your return was rejected because you forgot to check some box or left something blank. In that case simply correct the issue and e-file again. (I mention this so you allow a couple of days of buffer before the due date.)

Many states have similar online filing services, although some have restrictions with regard to the forms that are available (e.g. resident vs nonresident).

In your case the disadvantage is that you have to type in all the data from the forms you already have, so you are trading labor for cost savings and speed.

If you are worried about the speed of the postal service and your are expecting a refund (if you owe, the IRS might charge late fees and interest), you could also file an extension electronically (with minimal typing), so it won't matter if the return arrives a couple of days late. EDIT: As @DilipSarwate mentioned, you have an automatic extension until June 15 anyway, so the above only applies if you needed more time than that.

I successfully filed Federal, NY and NJ taxes from overseas this year. I just went to my regular post office and posted them like I would any other normal international mail. It took around 2 months from posting to receiving my refund.

Federal:

Department of the Treasury
Internal Revenue Service
Austin, TX 73301-0215 USA

New Jersey:

State of New Jersey
Division of Taxation
Revenue Processing Center
P.O. Box 555
Trenton, NJ 08647-0555

New York:

State Processing Center
P.O. Box 61000
Albany, NY 12261-0001

Although you are not filing a NYS refund, I want to add this in case someone else who is filing NYS from overseas finds this answer. They may note that the first page of your NYS refund has this big scary message:

This message is pretty much rubbish. I have never once filed my NYS tax return electronically due to my wife not having a social security number, which immediately excludes you from e-filing. Also they do not allow e-filing when you have an overseas address. You can safely ignore this message.