As to the .035 being too big and contributing to burn through... well... no and no.
kind of a 2 part there, I'll try to explain.
.035 flux core wire is quite a bit different than .035 solid wire. Solid .035 wire just has more metal in it, .035 flux core is probably more comparable to .023 solid as far as the amount of filler metal in the wire per inch.
So your .035 flux wasn't "too big" for the job. It just sounds like you were using too much heat, or zapped a thin spot (possibly ground too thin, possibly rusted thin from the opposite side) or you are just staying on the trigger too long in one spot.
Flux core will work, I just don't care for it personally.
As to part 2 (my second "no") even if it was .035 solid I’d have to say that the wire still isn't "too big" You can do a lot with .035 wire on sheet metal by just using less wire speed and less heat.
Personally I do prefer .023 for sheet metal welding, I feel that it lends itself better to the low heat settings with a quicker arc to puddle time when you hit the trigger. (it just starts a little faster) The .035 wire will go from initial arc to puddle at low heat settings fairly well but I find it a bit more clumsy than the .023 in this respect.
Run your heat as low as you can for your sheet metal to sheet metal welds. Run a little hotter for sheet metal to structure welds (plug welding to simulate/replace factory spot welds)
Start with floor repairs if that is part of your project. You get to practice on an area that will be covered up with a carpet lol! Get your initial experience with areas like that before you jump into body panel patching or partial replacement.
Burning holes and "burning back" usually means not enough wire speed. If you feel the wire pushing and popping/sputtering you may be trying to feed too much wire for the heat setting you are using.
As others have said, do NOT try to run constant beads of weld on body panel areas, you will warp the hell out of the metal. Tack weld in areas like that, tack and move, tack and move. Keep landing tack welds centered between tacks you previously placed until you have eliminated all the space between the tacks. The end result might not be as pretty as a steady weld but it will greatly reduce your warping.
here's a quarter skin going on, tack..
tack, tack..
tack, tack, tack..
Eventually it all becomes one weld. Carefully grind the area and it's ready for some filler to conceal the seam and help blend out the minor warping you will get even with the tack-tack method.