What I would do, even before trying any rust remover solution, would be to scrape and/ or wire brush the various problem areas, to try to determine the worst spot, as far as depth of rust and pitting. Then concentrate on that spot with scraping, picking, and wire brushing to see if you can get it adequately to clean metal, without actually going all the way through the tubing. If you do get a hole, well, no need to worry about the rest of the rusty spots, imo.
I bought a Trek 760 off CL several years ago, for $100, that didn't look too bad mainly rust/ bubbling paint around the top tube cable braze ons, and some spots under the top tube. Upon cleaning the rust, scraping & using pointed awl to get to the bottom of the rust, I discovered it went all the way through the tube in spots. Not trying to discourage you, just saying it would be a good idea to look at it really close before investing too much time into it. Best of luck!