HRM-Dual sensor - always on

No, mine does not stay active.  Make sure it is unsnapped from the strap or that could cause it to continue to run.

That would probably be the dumbest solution there could be. For the last 5 years on the previous sensor, all I had to do was remove it from the body (without unhooking it from the belt) and everything was ok.
The manual for it says that the sensor should last about 3.5 years on a single battery with daily 1h use. In my case, I use about 3h a day, so the battery should last about a year. A year versus two-three months is a big difference.

Is the strap still sweaty after your workout? Sometimes Garmin heart rate monitors will fail to turn off if a little bit of electrical current is able to leak between the electrodes. You can try wiping it off with a dry towel.

Hence my suggestion.  Garmin recommends rinsing the chest strap in warm water after every ride (after unsnapping the HRM from the strap) to prevent the current leakage.

I checked a few things and the case is as follows:
- the sensor stops working after at least 10-12h left on the belt.
- when i bought it i used it without any problem (that is, on the factory battery) from august 9, 2021 to april 7, 2022 ( i drive practically every day - in winter too :) well maybe except for the time when i am on skis ;) ) - that's a lot of time. The problem appeared after the first battery replacement. I use decent CR2032 - usually Energizer - not any no-name. 
- The suggestion to rinse the belt in warm water after riding - well average idea to tell you the truth - sometimes I have a break of several hours between rides - it won't have time to dry. Wearing a wet belt when you go cycling in the winter is already a bad idea in general.

- From April to today is the third battery....

At the moment I detach one clip - on average I like it, but oh well. If you do not have what you like, you like what you have :)