You've come to the right place.
I've been going to Mexico Beach two to three times a year for 45 years and my wife's family has a house down there. It is my happy place. Great place to do what you are planning.
Mexico Beach has almost new ramps given to the city by St. Joe Paper. They used to be free but now they charge minimally for use. You have to be able to raise your mast on the water because the ramps are on the North side of the canal bridge, but you can tie up to bulkhead and do that with calm conditions. From there two great places to go are Crooked Island/Wild Goose Bay to the west and the Cape San Blas point to the east. Both are very protected for overnights and are usually not in use much in the spring. Both these sails are about five miles total and you can stay very close inshore to make them.
Another option is using the ramp in Port St. Joe. From there you can sail the bay west to Cape San Blas point, or east to the Blacks Island area at the head of the bay. You can sail to the St Josephs St Park and use the park facilities.
From Applalach you can go through the bay to Indian Pass and I think a 16 can get through there, but it is a longish sail around Cape San Blas to the protected waters. I haven't spent as much time there, but I'm sure there are options for overnight without even leaving the bay. Think there is a State Park on the west end of St. George.
With regard to conditions, water stays cold until about mid May and the early Spring is liable to be fairly cool, with gusty winds. Staying in the Bay may be the best option depending on weather window.
When you get into the planning stages for this excursion, post or PM me with specific questions, will be glad to help.
Picture of the Canal at Mexico Beach. The Dockage here is fairly cheap, I usually dock my boat here when I've towed it down and done day trips from here.