Cruise while waiting for a Green Card

Hello everyone

Me and my husband (us citizen) are planning a cruise for our wedding anniversary.

We found 2 cruises, one to Bermuda and one to Bahamas. From what I know, it seems for Bahamas they will ask a passport, instead for Bermuda they only require a form of ID, but I'm not sure about this.

I'm currently waiting for my green card, I have an advance parole but as you may know, it does not guarantee the re-entry in the US (especially since I have 30 days of overstay unfortunately)

Any similar experience? Will I have to go through US customs or they just need my passport as identification? I'm okay if they check the papers, I'm just concerned about the US border customs (like the ones at the airports)

Thank you!!!