Tm4c nested uart interrupts- does each com finishes it's run before the next one executed?
Your problem will be solved if you give priority levels to your interrupts, so first won't bother second.

Here is one example:

NVIC_SetPriority(UART3_IRQn, 0);
NVIC_SetPriority(UART7_IRQn, 1);  

While one interrupt is being serviced, second arrived is in pending state.