I have a similar issue and mine is left plugged in most of the time as well.
I just pulled the battery on my 8800L and left it outside in the cool breeze for a half an hour. It's back on Verizon 4G. I'll have to see if keeping it off the charger periodically will work as a solution.
I still don't think it's unreasonable to expect a hotspot like this to be plugged in 24/7 (my city's EMS has Verizon hotspots like these in their trucks which are left plugged in) and it seems pretty unacceptable to have to deal with an issue like this. Thanks for the idea @Kristi46773! Hopefully this will work as a stop-gap until I can bring it in to Verizon.