I cruised pregnant twice late last year at 6 and 17 weeks. It was fun, but I was a little queasy most of the time. I was queasy on land, too, so might as well be queasy on a cruise!
They didn't ask for a note, or even how far along I was, and I was visibly pregnant at 17 weeks (hello, third pregnancy). Somewhere along the line I clicked a box online that said I wasn't entering my 24th week at sailing, so they take your word for it.
There are lots of "rules" for eating and drinking while pregnant, so you just need to decide which ones you feel matter. This was my third pregnancy, so I have done plenty of research on the actual risks. I ate a turkey sandwich most nights for dinner from room service because it was all I could stand to eat. Yes, there is a slight risk of listeria from lunch meat, but I had to decide: starve or "risk" it. Most nights in the dining room I requested potatoes for dinner. That was all I could handle in the MDRs. But our server was great. By night 4 or so of our 7-night, he was bringing me plates of whatever potatoes were on the menu that night

But by 10pm, I was back to starving and ordered a sandwich.
If you are cruising out of port Canaveral, the last night is the worst for motion. If you have morning sickness, be careful what you eat on the last day, no matter how you feel during the day. A hamburger was not a wise choice for me. Enough said.
On a related note, know where the bathrooms are near each restaurant. Food smells were the worst for me. Again, enough said.
We love Mixology and had a group planned from before I found out I was pregnant, so I just paid and went along. I would have one taste of the drinks I wanted to try and then give them away. If you want to make friends fast, that's the way to do it
If you have any specific questions, let me know. Despite the morning sickness, I still loved cruising, and I hope you do too!