Cost, time, quality - pick two.
Since you have a fixed, low budget, you'll either have to spend a lot of time scouring garage sales, classified ads and Craigslist to get the perfect bike, or else you'll have to settle for something that is close, but is not quite ideal. Can you increase your budget by saving somewhere else, perhaps by moving components over from your Fuji, or by keeping the old components attached to the frame you buy, and replacing them gradually as they wear out? Can you get another $60 by selling your Fuji?
My beater commuter bike is an early 80's Bianchi road bike, steel frame. 27x 1 1/4 tires. 1.25 inches = 32 mm. I believe that was a fairly common tire size in the 70's and early 80's. I agree with you about 23 mm tires for commuting though - my new road bike got 2 pinch flats in my last 4 rides, but I'm not sure whether I'll be able to even fit 28mm tires on it.