Is it possible to file for divorce without giving your home address to the person you are divorcing?

Depends on your reasoning and depends on if there are kids involved. You can have all correspondence go to your lawyer if you choose and that is the address that will appear on your dissolution order (at least in FL)

Is it possible to file for divorce without giving your home address to the person you are divorcing?

It is. You can use a P.O. box if you are self-represented, or say your address is confidential on some forms. You still need some address to receive mail from the court.

This is actually done a lot. Frequently, for safety reasons, the address of one of the parties is not listed on any documents. You simply need to make that clear to your attorney and all paperwork that he/she files should have your alternative address listed instead. If discovery is done by the other party and they ask for your current home address, you will still not have to answer that question with a direct response. Instead - your attorney will do the wording so that the discovery is still answered but your privacy is maintained.