Do I need to replace my rims to move up to a larger width tire?
You're in a bit a predicament because of the age of your bicycle. If I remember correctly, your rims are S6 (Schwinn's designation), and therefore are ISO-597mm. This size was used by Schwinn are their mid-range to high-end bicycles in the 60s and 70s, and on a few British bikes. It is not compatible with nearly every modern tire. However, Kenda does make a 1-3/8" tire in that size. It's not bigger, but it could be a viable replacement when they inevitably need to be.

If you want a better selection of tires to choose from, I'd recommend taking it to your local bike shop. They'll be able to assess how you'd like to proceed. You'd likely be replacing your rims, and possibly the whole wheels. The prices could range widely and will depend on your area, so I don't want to comment on that.