MM,
SHNF is big, so there is no 'best place.' You also do not have to worry about shooting a 'big ass' buck.
Sounds like you need some help, or you will be in over your head. There is a hunting license and a public hunting permit to purchase. There will be rules and regulations for Texas, and special rules for NF lands. If possible I would (based on your posts) find an experienced friend to hunt with.
Bow season is October. Gun season is Nov-Dec, with bits and pieces of Texas extending into later/other times.
My own experience in SHNF is disappointing. I drove toward what looked like a public landfill, only to find out that it was trash left by "hunters" from opening day at a designated campsite. I was there mid-week, but still saw many others decked out in blaze orange. Also, although it says 'National Forest' it is festooned with roads, trailers, and houses. Be very sure of your location and what lies beyond.
The best hunting I have had on public land in Texas is during bow season, because there were so few others in the woods/swamps. I would not recommend early November, due to the crowded conditions. Wait for bad weather and hunt in the rain. If it drops below freezing there will not be many others hunting, relatively speaking. Black powder also has some separate seasons which might help thin out the hunters.
Your best bet for quality public hunting is to pay $$$ to get in on the drawings. Anywhere in Texas (mostly west of IH-35) that is public land and has good hunting has been put into the pay/drawing category. If you purchase an Annual Public Hunting Permit you will get a booklet with the info.
You might also look at feral hogs in the eastern forests outside of deer season. Or make friends with someone who owns land...