How friendly are you with your dealer? It's not unheard of for Stihl to cover the cost of parts for a failure like this that's technically out of warranty but clearly a manufacturing issue. Just takes a little extra effort/phone call from your dealer usually.
It is not the module; Stihl calls it a solenoid. I have had the carb apart twice. I left the air filter and the piece the filter attaches to confirm what the saw is doing and sure enough it is puking huge amounts of fuel out of the carb. At this point lacking the software and ability to plug into the diagnostics port I have thrown in the towel on it and am going to take it to the dealer which irks me to no end. It is a lot like when the ABS on my 3500 crapped out and I knew one of the three sensors was bad, but a standard code reader won't pull the code since it is a separate system, so had to pay someone to fix it. At this point I think it is the coil pack, but thinking and knowing....
So far the only other M-Tronic saw that I've known of to have issues.......was also a 241. It had a bad coil......and just quit running.
Foragefarmer, did you resolve the hesitation issue & if so what was the fix? I picked up a MS241 a couple weeks ago and through the first 2 tanks it's doing what you described, almost like it's flooding out. Idle time between cuts doesn't matter, if the chain stops it is going to bog & you have to gradually throttle up to make the next cut. I tried the 90/90/5 cut thing mentioned on the 1st page & no change. It's back at the dealer now.
The dealer for sure will give a solution just because the only think that they know to do is to change new parts with other new parts , even if only a untight screw is the solution of the problem . STIHL thank you for your support .........................