I had 100+ dives in more than a decade blowing bubbles, before I went on my first drift dive in the Yucatan several years ago, and I also had a bit of anxiety about it too. But I found it enjoyable.
Boulderjohn pretty much covered the answers, but I will add:
Before you splash in, make sure you are all squared away and ready to descend. It's not likely you will be doing a negative "hot drop", but you still dont want to be fidgeting with your gear on the surface. When the DM says "drop" be ready to drop.
Buy the SMB, and practice using it. If all goes well, you will ascend with the DM, and they will deploy the SMB. However, there may be instances where you will need to deploy the SMB yourself. If you get blown off on your own for example.
Before I went on vacation, I practiced using a SMB in my cold home waters, with mixed success and sometimes laughable failure. On the drift dive, I asked the DM if I could deploy my SMB and I was successful - we used it to surface. A small point of pride for me.