Is your unit very old? It could be the internal battery. LiOn batteries only last for a certain number of charges and then must be replaced. You can buy replacement batteries and install them yourself. You'll need a special tool to get the back off the unit. I've looked at the replacement battery for ours and it's a simple plug-in battery, no soldering required. You will, however, void the warranty if you happen to still have one. But if you have a perfectly good unit, why not try a new battery.
Or maybe this is a good excuse to get a new model.
Still sounds like it could be the battery. I have a Sirius XM radio doing something very similar and I'm pretty sure it's the battery as it's a 2007 radio. I just haven't gotten around to changing the battery yet. Very old batteries that are completely toast will not even accept a charge at all.
Garmin will not continue to charge
