If you rotate the tires while it's off the ground, does it feel like there is a lot of friction?

It might be that holding it in the air relieves enough stress that it appears to be spinning up to speed.

Batteries can still read regular amounts of voltage and still be dead. It sounds like this could be the case. They might read the 4.8 volts but still not have any capacity left to supply the needed amount of current.

Driving really slow doesn't sound like a problem with the remote. I would looke at the batteries and the friction of the drivetrain. Check your connections to the motor too.