Will you have a car for that period of time? If not, I recommend HCR. There's about 500 bolted routes in the ranch, so you really shouldn't run out of things to climb. If you do have a car, then Foster Falls is a good option. Foster's doesn't have as many routes as HCR, but the routes are generally longer, and you'll be very close to plenty of other crags of various styles and quantity of routes (like Obed).
For places to stay, you can rent the cabins at HCR. I don't remember the cost, but it is cheaper during the off season for the dude ranch part of HCR. The cabins have AC, heat, a fire place, tiny fridge, microwave, coffee pot, and full bath. There are also several cabins nearby on the highway to the ranch, and I think one even off the gravel road on the way in. Worst case, you could rent a motel just down the hill.
If you're not going to camp at Foster's, I highly recommend The Crashpad hostel in Chattanooga. It's about $30 a night for the basic bed, but this place is really, really nice. There's a kitchen available, they offer breakfast foods, the beds are really private, everyone there is super friendly and almost surely a climber or other kind of outdoor enthusiast. And it's conveniently located in downtown Chattanooga within walking distance of anything you could need. This place is very clean, and quite unlike what I think of when I think of hostels.
Mid November is jacket weather for me in both locations, but I'm from south Louisiana so take that for what it's worth. Around that time of year I'll might wear long sleeves, definitely a light puffy or fleece, and maybe thermals if the sun's not out, but I find it to be still very comfortable weather to climb in. I see you're from Denver; don't worry about the weather, you'll be fine. It's probably short sleeve weather for you!
You didn't mention trad climbing, but both areas have some very fine southeast sandstone trad crags. Only 45 minutes from HCR is Sam's Throne which has the typical turtle shell/dinner plate style trad routes of the southeast and some harder lines, too. There are some neat crags very close to Sam's Throne (Cowell, Haw Creek, etc.), and all of those could keep you quite occupied for the time you're there. Outside Foster's you have Tennessee Wall, LEDA, and Sunset Park. I find the trad climbing to be more straight forward in Tennessee compared to Arkansas, as in the cracks are more continuous and the lines a bit more obvious, but I love the adventurous nature of trad climbing in Arkansas.
If I could surmise both areas, I'd say climbing at HCR is more relaxed, with the average route length being about 60 feet (don't quote me on that, that's what it feels like to me), the texture and holds being easier on the fingers than Foster's, and lots to choose from for anyone climbing 5.6/5.7 to 5.12. It's not too hard to satisfy a group of climbers of different strengths in a day since you're generally not too far from something easier or harder, and lots of times you can find walls with everything from 5.7 to 5.12 in the same wall! I think the grades are a bit stiffer at Foster's, but the routes are also longer (can be, there are some sort routes, too) and the nature of the rock is different as well. The rock just feels older to me at HCR, and fresher at Foster's.
I'm not a strong climber, I hate hiking over anything other than perfectly flat trails, and I prefer vertical walls to overhanging ones, so my preference is Horseshoe over Foster's. Both are good climbing destinations, but if I had to choose where to spend 5 or 6 days it would be HCR.