This would be my recommendation.
For the paddock I’ve used a small tent like this for years and it’s been great. I drag it out in the summer as well just to have a shady spot to relax between sessions. With all my gear in it I’ve only had one or two days where wind was a concern.
I’ve used one like this for several years now, rain or shine. It’s large enough for a low chair and all of my track gear; tools, cooler, jack, stands, GoPro bag, etc.
One thing to watch for with this style sunshade is many of them have “windows” with the flap on the inside. So they aren’t really weatherproof as the rain can just drip inside. This one doesn’t have any windows and zips completely up.
I think I’ve only had two days out of the last 5 years where it was too windy to get it set up. Once all of my crap is loaded into it, it’d be pretty difficult to blow away.
https://www.amazon.com/Coleman-2000002120-Beach-Shade-Green/dp/B0009PUTIE/
Well if you are going to need to walk a block or so to the beach, I recommend a jogging stroller (they are easy to push or pull on the sand, and also keeps baby out of sand if needed). I also recommend a canopy like this for nap time. Keep baby in a rash guard type top with bathing suit. If going in the water, get swim diapers, if not getting too wet, regular diapers. As /u/da7rutrak stated, baby powder will remove sand, bring excess. If you are weary of the ocean, you can buy a cheap throw away when you are done kiddie pool and a bucket to put ocean water in said pool.
The drive is on you, stop frequently to nurse / change diapers...you don't want baby getting a rash, that will make the trip so much worse. Best of luck!
Edit: forgot a baby hat (with straps to keep baby from taking it off) and lots of sun screen, remember to put it on about every hour.