Advice from a northerner - use with caution.
IF the wheat is thigh high or will be, there is a lot of straw out there. There is also a lot of grain out there, most likely.
I think the best use of it would be to leave it stand until after 6/1. The heads will mature and grain will fall down reseeding itself.
After most of the grain heads have opened and the seed fallen to the ground, have someone cut andbale the straw for you and stack it by the road. Someone will want to buy it for bedding, mulch, etc and you might want to keep some bales for yourself for miscellaneous use. Highway crews love wheat straw to blow onto new seedings of grass after road work.
Put a small ad in the paper or call the local highway dept and ask for names of who they or contractors use to reseed road work. They will pay a good price for nice clean, low-weed seed content wheat straw.
Mean while the grain/seed will reseed itself and as mentioned above you will hav some very nice fall wheat to graze. In the fall after grazing the wheat down, no-till drill a pasture mix of locally adapted perennial grasses hopefully with a good percent of clover. I'm not sure of the best timing for drilling in the fall in the south. Maybe others here can fill in on timing.
Wheat does reseed itself and make good grazing. I would NOT burn it off nor use a herbicide. I would get much of that heavy straw off of there by baling so the reseed wheat and the no till drilled grass seed can emerge.
I would either rent a drill or have a neighbor with a no till drill seed the grasses. The reason to drill rather than work the ground is so that you don't lose the reseeded wheat. The grasses/clover can emerge nicely thru the baled wheat stubble.
Sounds like you have a very nice arrangement there. Best of luck to you!