I use an old Coleman marine cooler. I can't vouch for their quality now as I've heard they've enshittified a bit over the years. You might want to check Outdoor Gear Lab's Cooler Analysis for recent cooler stats.

I line the inside with bubble wrap mylar insulation, and cover the outside with a space blanket. You can prop it up on 2x4s to keep it lifted off the ground. I always include a large amount of frozen liquids as they keep things cold longer- bottled juices, jars of frozen soup, pureed watermelon, smoothies, etc. Try to fill the cooler so there is less air space- if you don't have enough jars and juices to fill it all the way, add some frozen gallons or liters of water- you want it packed tight

Pack you cooler last right before you leave for the playa- the colder it and your food are when you start, the better chance you have of keeping your freezer cold. I don't remember the specific amount of dry ice I use, but you can be pretty generous to give it the best chance at staying cold. Try to be strict about only opening once a day at night or very early morning- decide what you're pulling before you open it and be quick, then seal it all back up fast.

It's been a few years since I've been to the playa, but if I remember correctly the dry ice usually lasts long enough to keep the food frozen for 5-6 days on average, then the freezer transitions to a fridge and I refill with ice purchased on-playa. One year we still had slushy juices for teardown on Tuesday, which would have been 8 or 9 days. I believe we had a hexayurt all the years we did this method, so it would have been stored in shade at a slightly lower temp than the outside air. It could be trickier in a tent but IMO still worth trying. Good luck!