Some great suggestions here!
I’m a newbie TIG welder too (actually, newbie ‘welder’ full stop). But I have experienced almost all of your issues and I am struggling through.
I have found that I can reduce burn through by laying a 1/16” filler rod right down on the buttweld seam. Melting the filler and spreading it out on to the two sides of the base metal. While welding really thin sheet metal (like 20 ga), I go from seeing a puddle to an eighth inch hole in a nanosecond. The added mass of the filler material gives me a chance to actually work the puddle before burn through.
The second thing I’ve found helpful is to start my arc a little to the side of the buttweld seam, and work my way to the seam as I add filler. When I try to start the weld right in the seam, I often just evaporate the edge of the sheet metal, making the gap wider like what I see in your picture.
Third, I’ve had quite a bit of success doing a lap joint, but using only a very short overlap (<1/8”). I’ve found I can melt the upper layer fully into the lower layer, eliminating the lap joint and ending up with what looks like a very nice joint.
I have some rust repair to do. I’ve been debating about using TIG, but I’m almost positive I’ll be using MIG. I am way more confident I’ll get a nice product with minimal warping by stitching away with the MIG. Especially when I go from feeling almost competent doing the TIG in the Horizontal, to trying the same in the vertical…