The puncture pin shouldn't make contact at all with the can (even after fully screwed onto the line) until you screw it further in to allow it to make contact/puncture. I would say you're doing something wrong. All lines should be connected to the gauges and system with the valves closed. Then connect your freon can to the fill line. Then turn the handle to push the puncture pin into the can. Only then, should you open your gauge low pressure valve and fill the system. There should be absolutely no freon lossed to open air.
Your AC guy should have added oil to the system when he worked on it. Especially for $700. If he didn't, and still wants another $900 to fix what he should have already fixed, I'd find another AC guy (at a minimum) or better yet, do the work yourself. Labor is cheaper that way and it's really not hard to work on these things.
And most of the time, "Stop Leak" stuff is only a cheap band-aid fix that never really fixes the underlying problem/issues. Some people get lucky with it, but most of the time it starts leakying again or messes something up really bad. (Same way with radiators...)