The unit I have keeps swithing off even when fully charged. What could be causing this?

If booster is fully charged and the vehicle battery is almost discharged it is inevitable some battery charging will take place but it is likely to be a slow and incomplete process and could shorten the life of the booster which isn't built for such work.

It is also worth noting the car alternator is also not built for charging the battery and is often incapable of doing so. The purpose of the car alternator is to keep a fully charged battery fully charged and a discharged battery can even cause damage to an alternator.

All I can suggest, as you are doing , keep trying to contact service provider , as your problem need their help. Is their a RE SET button on the unit.

Replace the inner part of the fan - some unit just slip right out.
check on kijiji and find a local computer repair guy that will sell you one for 5 or 10 bucks, or get a broken laptop for 50 to 100 and lots of parts. I had one broken wing and broke on off the other side to even it out... even though one side had 1 extra wing it wasn't that far off balance so vibration should be minimal.
if you keep your laptop clean inside around the heat-sink and fan area and put silver heatsink on the gpu and cpu as well as install the manufacturer drivers and update them to cool things down and speed things up and keep things from heating up.
One more issue is the charging port and charger end get frayed and or loose and will cause heating issues and have caused fires.
Replace batteries that are weak as they are continually charging too much.... Store batteries and at 70% and don't pull them out of the laptop while it's plugged in.
Fully charge batteries and fully discharge them between charges.
Good to have 2 batteries.
Don't use hot batteries especially after charging and don't charge a hot battery.

aka Mr.Fix-it!
East Coast Canada

it sounds like the starter motor or the earth lead to the engine could be broken off or loose

You could try say Ebay... or the like.
You cannot usually charge a totally flat, or even a half flat battery from the USB port. These are on;y used to "Top UP" the battery... You need to replace it with a fully charged one,and then us the USB to keep it fully charged.

A few things that cause drop out.

Poor vicinity for a strong signal
A low battery ***** must be fully charged The DC is only maintenance charging and eventually goes flat.
A damaged SIIRF CHIP caused by starting the engine whilst the GPS is plugged in.
A failked fuse to keep battery charged for maintenance in the DC charge unit.

Try recharging battery( or replacing it if 2 years or more of age)

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If you have another power charger try it. It can only be one of two things. Bad charger or bad battery. Most likely the battery will need replaced

Locking the player causes it NOT to charge.

IS the battery charged fully? Try turning it on while its plugged in.

Batteries have a limited life even if they are constantly charged.
Leaving batteries in constant charge will limit their life by causing gas leakage. Once depleted, they must be replaced. There is also a small button battery that keeps unit memory up, if you do not use camera for months at a time, replace this battery also.
Ni-Cad batteries need to be refreshed ony once a month by discharging them fully, then recharging them fully for best performance to erase memory effect