I'm seeing ZAMA C1Q-S211 STIHL 1141 120 0606 as a possible carburetor for the MS261 and it looks like it has the same physical oddities.
It looks like your good friends in China will deliver one to your mailbox for $19
https://www.ebay.com/itm/Carburetor...996586&hash=item1a7c9806fb:g:S7IAAOSwsxJa7WZf
Before ordering one, I'd pop the cover over the diaphragm and see if your saw has has a drinking(ethanol) problem.
A solution to your problem could be as simple as a partial rebuild of the existing carburetor.
This is super easy to do, and even a picture of the diaphragm would help determine if that is the issue.
It looks like your good friends in China will deliver one to your mailbox for $19
https://www.ebay.com/itm/Carburetor...996586&hash=item1a7c9806fb:g:S7IAAOSwsxJa7WZf
Before ordering one, I'd pop the cover over the diaphragm and see if your saw has has a drinking(ethanol) problem.
A solution to your problem could be as simple as a partial rebuild of the existing carburetor.
This is super easy to do, and even a picture of the diaphragm would help determine if that is the issue.