I am not sure exactly what would be causing that problem. Since the batteries were recently replaced I would check to make sure that no connectors are loose or unplugged and that all of the wiring is intact. Batteries are very heavy and installing them could accidentally pull on any wires and connectors that are in the way.
In the mean time I would not worry too much about the turn signals draining the batteries. Turn signals draw a very small amount of power and the batteries are very large. I would go ahead and use it as normal until the turn signal problem is solved.
The tech at pride mobility dealer installed new batteries in wife's three wheeled Maxima scooter. now the amber / turn signal lights come on when the key is turned to the "ON" position. when the individual urn signal switch is set, the turn signals flash as normal. when switch is set to off, only the right amber light goes off as set. left light will not go off until the key is turned off. wife called the repair tech, he stated all he did was change out the batteries. but wife found small washers on deck of scooter,way to small for battery connections. now going on two weeks waiting on scooter tech to return calls.wife won't drive scooter, afraid of running out of power because lights are draining batteries.any ideas or help would be helpful. can't say much for their customer service at this point.