While the transmitter is a common point, unless a setting has been accidentally tweaked, it is highly unlikely to be the culprit. Digital devices work fully until they don't, and then they don't work at all. Generally.
The common item with the batteries is the charger (apart from their age). A smart charger that runs into "unexpected conditions" will often claim to have done its job when in fact it hasn't. A session using a dumb charger and a meter can tell whether things are working as they should.
Roycv's check will help. Check the voltage as charged, under load, and again after use under load. A nominal 12 volt battery that drops below 13 volts under anything but extended use is suspect, but only after charging with a "dumb" charger.
The control pot in an old transmitter will wear, but I would not expect to see the range of control reduce. Spluttery, patchy performance, yes, a worn track or wiper will do that for you.