I'm pretty sure you can choose not to see these Twitter DMs in your general inbox. They come through mine as DM requests and I can view them before accepting the message and interacting with the freeloader sending them to me. (They are always freeloaders). Is that how it's working on your end or are they all flooding into your inbox?
That’s how it is on mine too, the reason I always reply at first is because I want to be accessible and maybe turn them into clients....guess that’s just a pipe dream though. Lol