Radio acting up

Blake.
For practical purposes, all of our glider batteries are deep cycle batteries. The deep cycle batteries are intended for moderate currents over long periods of time (say 1 Amp over 6 hours for a 7Ah Lead Acid AGM battery). Contrary to a car battery that is intended for very high currents (hundreds of Amps) for a very short period of time (seconds that the starter is running).

Your problem description above is very limited, so there is a lot of guess work going on in the responses. You will have to measure to debug your system.
In my experience, most radio problems are in the wiring. First check that your wiring is solid.
Then check the easy things.
Is your PTT button solid or some flakiness there ?
Does the transmit indicator on the display stay solid on while you are transmitting ?

Does the radio display start blinking when you hold the transmit button down for 5 or 10 seconds ? The supply voltage is dropping below 10.5V. That would be an indication of a power supply problem. Could be a battery problem or a wiring problem.
Measure the supply voltage, preferably at the radio or at the panel distribution point if your battery is remote (like behind the seat). Push and hold the PTT button for 10 seconds and see if the voltage remains fairly stable.. Should not drop more than 0.5V.
If it drops significantly, check your battery for the remaining capacity. You cannot determine the battery capacity by only measuring voltage. You need to fully charge the battery and then run a load (e.g. car brake bulb) for hours until the voltage drops to 10.5V.
Product of time (in hours) and average current (in Amps) is your remaining capacity.

When your buddies receive your seemingly interrupted transmission, what exactly do they hear. When your modulation quits, do they still get the carrier (i.e. silence) or do they hear noise in between your words (i.e. the carrier is switching off).
Test it with a handheld with the squelch off to avoid the squelch from suppressing the noise.
If the carrier is interrupting, it might again be a power supply problem like above.
If the carrier remains on, it is pointing to a microphone or a radio electronics (modulator) problem.

Hope this helps. Good luck with your troubleshooting.
3U