Chainsaw Repair

In the last 30 years in the trees I have dropped a saw out of a tree twice.
First time was my 038AV in 1986, before learning to make my top cut 'behind' the bottom cut on a heavy horizontal limb.
It was the classic 'broke away and took the saw with it'.
The fall broke a small piece off the side cover, and yet it still runs well to this day.

About a year ago I took down a very large five-leader cherry.
As I climbed up through a lot of secondary growth on a leader that had previously lost the top in a storm
a small branch twisted open my snap and my 346xpg fell about 40 feet ('now' I have a saw lanyard).
It landed vertically on the back handle and trigger parts flew off in a couple directions, also cracking the gas tank.

I bought a used Husky 350 inexpensively that had a scored up piston/cylinder and used the gas tank/rear handle assembly to rebuild the saw.
Ever since it runs until it gets warm and then craps out. No kinks or leaks in the lines (impulse or fuel), and good compression so I don't think it is a crank seal.
I even paid the local shop to have a go at it and they gave it two goings over before crying uncle. Carb kit, new lines, no better
I am using my backup 346xp, but miss the heated handles this time of year.

I have rebuilt a saw or two over the years but by no means am I anywhere near an expert saw mechanic.
Any ideas what else to look at?

Has a spark as it still runs, just acts strange; poor power and then clears up for a cut or two and then runs strong; then hard to start, then okay, real inconsistent...

I was thinking coil.

I had a saw that would run perfect till it got good and hot. Then shut down or on  refuel would not fire back up.

Check the tank vent! The 350's are notorious for getting plugged tank vents and vapor locking. Cut should have an IPL, it's a bit of a bear to get to, but about a $3 fix.