Using Character Match functionality in a STM32F7 chip
The DMA CNDTR register counts down to zero after each transfer and shows how many elements remain to be trasferred. You can use it to check whether the DMA is running or not.

Your UART interrupts, if enabled, will always run alongside the DMA transfers and interrupts so the UART interrupts will have to chech the DMA's CNDTR register to know whether or not to actually execute.

You can also disable interrupts from within interrupts. So your DMA transfer complete interrupt can disable your character receive interrupt (or vice versa) and you can re-enable it elsewhere. Remember you also have interrupt priorities to handle when the last DMA transfer is "\0" character and triggers both the DMA transfer complete and character match interrupts at the same time.

In my code, I use character match to detect the enter key to know that a user command has been entered and to parse the message in memory being stored by the DMA. I also have a UART receive interrupt to echo characters that echoes the character directly from the UART receive register. These all run at the same time along with the DMA transfers and interrupts.