IMHO, the most likely explanation is that the battery is failing. Most Li-ion batteries seem to die at about 3 years, which I'm guessing is how old your One XL is.
Replacing the battery isn't easy, but can be done. See: TomTom ONE Battery Replacement
I know the battery is good because I replaced the old one with a brand new one, and the problem persisted.
New batteries come partly loaded, this reserve is almost used by now, and new juice is not coming in
This also means that the charger does not feed the running TomTom itself, as it would normally do.