Is Sunday a safe, non-hunting day in Wharton to have kids out exploring without having to be worried about wandering into something?
Sunday hunting is for archery only and only on WMA's and private property. You would be safe, even on a Saturday, year round in the State Forests in my opinion. Hunting is statistically one of the safest outdoor sports there is. I am in my 48th hunting season this year and I have never had an incident with a hunter or non-hunter.
The only week I facetiously warn people about is Six Day firearm for deer. That is December 5-10 this year. You will see a lot of people and deer clubs in the woods that week, especially on Saturday the 10th. The clubs drive deer and cover huge expanses of woods with their drives and there is a slight chance you could end up in the middle of a drive.
After that, there are several more gun seasons for deer such as muzzleloader and permit shotgun but they don't see the level of club participation that Six Day sees. Archery is open throughout the fall and into the winter in most zones. Then, from January 1 through March 15 you have predator season.
If you are not comfortable, I would recommend an orange hat and vest.