Do you have a bicycle?

Make sure the chain isn't too tight or too loose. There should only be a couple of inches of slack in the chain at the most (i.e., a couple inches of movement if you grab it and move it up and down), and about an inch of slack at the least. You can get away with less slack than that if neither your chain wheel nor sprocket are out-of-round in the slightest, but that's highly unlikely.

Also, check the alignment of the sprocket (small rear gear) and the chain wheel (larger front gear). This is a picture of my Mongoose which has perfect alignment:

https://i.imgur.com/aJcmbTp.jpeg

And you can check it with a straight-edge like this:

https://i.imgur.com/hQdyPcw.jpeg

If both gears don't simultaneously sit flush against the straight edge, that can cause the chain to fall off. Best case scenario for misalignment is if the sprocket is just a little cockeyed relative to the chain ring, because that can usually be fixed by repositioning the rear wheel within the dropouts. Worst case is if the sprocket is on a significantly different plane than the chain wheel. The only way to fix that is by adding or removing axle spacers, which might involve taking the hub apart (or adding or removing spacers between the bottom bracket and the chain wheel, but that only applies to certain types of crank sets).

"It's showing its age, but it still rides nicely so I see no reason not to keep it."

Definitely keep it. It will last forever as long as something drastic doesn't happen like it getting run over by a car, or being left outside for a long time to rust away. Those "Electro-forged" frames are very cool, and are unique to vintage Schwinns.

Bearings and cones/cups can wear out but those can easily be replaced. The biggest thing would be the coaster brake, which is a type of drum brake. If that wears out you'd probably need to find a complete replacement rear hub (for example - https://www.bicycleheaven.org/products/schwinn-bicycle-five-speed-road-bike-hub-others-1970) because I doubt you'd be able to find new internal parts to rebuild it. Most of the old/good ones were manufactured by Sturmey Archer or Bendix. In fact, Sturmey Archer still makes them, which is what I'd buy if I needed a replacement (because who knows what the internal condition of an old used original one might be?):

https://www.sturmey-archer.com/en/products/detail/s1c-silver

This is a good article about coaster brakes in general, including pros and cons:

https://www.sheldonbrown.com/coaster-brakes.html

"They did and I made a great first impression with her — as I rode up to them, I did a face plant off my bike right in front of her. Another reason why I don't like hand brakes, probably."

LOL. I love front brakes (they have more stopping power than rear brakes due to weight transferring to the front when you brake), but you do have to be careful with them. When I was a kid a lot of other kids were scared of using their front brakes because of what you described, and I was too when I got my first bike that had front brakes. But once I discovered that I could shift my weight rearward when applying the front brake and I'd never go over the handlebars no matter how hard I braked, I really liked how quickly I could stop with the front brake and it wasn't long before I was using it more than the rear brake. These days I almost always use both brakes simultaneously and shifting my weight rearward when I do it is second nature rather than something I have to consciously decide to do.