Thanks for the advice! I can't find any sharp bits inside and outside the tires/rim (I even sanded a few potential rough spots smooth), I have schrader valves so I don't think anything is pinching, and my rim tape is seated (the sticky kind) and covering the spoke holes. I have 27 by 1 1/4 inch rims, 27 by 1 1/4 inch tires, and I bought tubes that should be compatible (Kenda 700 c by 28-32 mm), but they did feel a bit small, diameter wise, as I was putting them in. People seem to say that 700c and 27 inch tubes should be interchangeable, but I noticed that some people sell tubes specifically for 27 inch wheels. Could that matter?

I did get a patch kit today, so I'll try patching my tubes and inflating slowly to check for problems. I'll let you know how it goes.