Unless you are measuring the voltage of your batteries under load, then your voltages do not tell you much. A battery that is just about dead will read it's nominal voltage.. until a load is applied and it can't keep up with the current draw...thats when the voltage drops. About the only thing you can do is get a meter that will read the current draw of the TX (like a "watts up" meter... a good investment if you want go get into higher powered electrics)... It shouldn't be more than 1 to 1.5 amps... and really it should be around .5 amps but some of the cheaper equipment seem to burn power for no good reason. I think those high power alkalines are good for like 2000+mah.. so if you burned them up in 5 minutes that means it is pulling over 20 amps.... I can't see a transmitter doing that not melting something...let alone work properly.
At this point I would say your best bet is to buy a package of batteries and put a full fresh set in and see how long it lasts.