Is Anti-Exploit module separate or part of existing shield (File System Shield) ?
Separate
Do I have to turn off the Avast Anti-Exploit module if I have turned on Malwarebytes Anti-Exploit?
No, you can have both. The technology of Avast Anti-Exploit is kind of similar to Behavioral Shield, but with focus on different characteristics (and different malware behavior). It runs on background and evaluates the processes behavior. It also scans the process for know virus signatures in certain trigger points.
Anyway, if you find some incompatibility with 3rd party products (like Malwarebytes Anti-Exploit), please let us known. Thank you.