I haven't had a 3G kindle since my keyboard model purchased in 2010 and long since passed on. But here's what I'd suggest, if you've not already tried it.
With the battery full and 3G turned on, do a hard restart. I found with the keyboard model that doing that would reset the modem when it was being glitchy.
With the battery full and 3G turned on, do a hard restart. I found with the keyboard model that doing that would reset the modem when it was being glitchy.