Ayleana,
Thanks for getting your kids outside and sleeping in the tent. I'm sure it helps them connect with the great outdoors. I'll try to answer your questions.
First, as to who you might possibly rent from, there are two primary companies it seems (although others can chime in if they wish). Note that the rental cost is in addition to the cost of a campsite.
The first company, Fort Camper Rental
http://fortcamperrental.com/ rents popups. These are the most affordable and offer a "tent like" experience. I own a popup and camp at the Fort almost annually and the connection to the outdoors is wonderful in a popup.
The second company, Greenberg RV Rental
http://www.greenbergrvflorida.com/ rents travel trailers and popups. Both vendors generally have good reps here and others are free to weigh in.
What time of year do you want to come? Summers have daily rain showers (June-late Sept) and I would suggest being an experienced camper if you tent during those times. It gets hot and humid too which can make sleeping uncomfortable. Some tenters bring an air conditioner for their tent during these times.
Winter time (late Dec to mid-Feb) can see 3 season weather. It can be really cold or sunny and pleasant. Shoulder seasons (spring and fall) are ideal tenting weather.
How much ancillary camping gear do you have? I mean, do you have a stove, pots/pans, awning if you were to tent camp? Could you handle heavy rains, cold temps, or hot nights?
The safe bet would be to rent a trailer or popup. Maybe given your comfort level and the kids comfort level with tenting, a popup might be a good start. They set up for you before you arrive and when you are done at the Fort you drive off and they take down. You can bring groceries or have services such as wegoshop or gardengrocer deliver groceries to your trailer that you have preselected.
As for DH being tall, some campers do have pretty big beds (my popup has a king). He could lay at a slight angle and have plenty of room. I've not heard of the straight wall tent but I have some EZ Ups I use when I tailgate at college football games and this year I bought some side panels to use. I put 4 on one EZ Up for one game and my wife commented how much it felt like a tent.
I would only invest in a tent solution if you felt like you would get a lot of use out of it and do more family camping. If backyard camping is all you'll do, I'd rent the camper.
Finally, straight out, with little kids IMO I'd do a rental and maybe as they were older go tenting at the Fort. But then again, they might be spoiled by a trailer.
Come back at us with more questions.
Bama Ed
PS - I have never used either company linked above. You can use the "Search This Forum" link to find other threads about them.