Can an upright bicycle be modified to deal with deteriorating balance?

If your balance might be an issue now, then a trike makes a lot of sense. The front end is like a regular bike, so you can have a basket. And the rear has a low platform or tray between the rear wheels for carrying other things.

I would not recommend any bolt-on stabiliser wheels (big training wheels) because 3 wheels can all touch the ground, whereas 4 wheels only touch the ground when its perfectly flat. So you can loose drive.

The "best" bike is the one you find comfortable and will ride. That's totally subjective, so you would be best-served to try out some bikes in person. Mail order doesn't tell the entire story.

One side note - some trikes have only one rear wheel connected to the pedals, and other designs have a rear differential to split power between both rear wheels. If you can test ride both sorts that may show some differences to you.

Lastly, electric assist is possible nowdays. The bonus, it saves your knees, but the electric bike should cost around double what the non-electric bike costs. Otherwise the bike parts have been cheapened to hit the price point.
Also, an electric bike can get you deeper into trouble, faster. Depends what level of "assist" you set.

In our small retirement subdivision there are a couple of ladies with recumbent trikes. These seem like an excellent choice (and I'm even considering getting one for myself).

The two main problems with them are that they take up a fair amount of garage space, and they are, at present (due to COVID) largely unobtainable.

I wish you well in your quest. Here are my thoughts, but keep in mind there is absolutely no right or wrong answer here. Just ideas.

I have seen several adult tricycles out in about. They fit like comfort bikes, or cruisers, but they won't tip over. You can look at some pics I found on this page: https://www.gearhungry.com/best-adult-tricycle/ This site links to amazon, but DO NOT BUY ONE ON AMAZON. This is for a lot of reasons, mostly you'll miss out on the help and expertise that you would get at a bike shop.

If you still feel like you will be OK on two wheels, there are a lot of manufactures that have some pretty nice "comfort bikes", and many have gears These put the rider in an upright position. Again, it would be worth a visit to a bike shop. The brands you have to choose from will depend on what your local shop carries, but pretty much all of the bike shops I have been to have something that I could work for you (not necessarily the tricycles, but they might.

As far as where to go... I can't say this enough. I strongly recommend a "LBS" (local bike shop). Not Walmart, or any other big box store. If you go in they'll let you test ride whatever they have and you can get a feel for whether or not you will feel comfortable on two wheels. If they don't sell they adult tricycles, they will probably at least be able to steer you in the right direction.

Lastly, the pro's and cons as I see them of a bicycle or a tricycle:

Bicycle: Pros...Lighter, easier to maneuver when storing in a garage or similar, easier to transport if you wanted to drive it somewhere but one significant con...no guaranteed stability. Only you can make that call.

Tricycle, pretty much the opposite. 1 major benefit. STABLE! with the the drawbacks that they are heavyier, hard to move around it tight spaces, require more storage space, and are difficult to transport.

Good luck.