I can help you out. I live in MO and hunt both private and public. Are you planning on hunting rifle or bow season? Bow season you get 2 tags and 2 turkey tags. As for public hunting like anywhere else rifle is going to be more crowded than archery and weekends will be more crowded than weekdays. You can go at different parts of archery(or muzzlelaoder) and thousands of acres to yourself. Also what time of the year are you coming. I suggest avoid the Oct lull, yes you can get some nice deer just like you can anywhere anytime, but statistically speaking it has a lower chance or harvesting a big buck. North of hwy 70 is what most people consider "big deer country" and for the most part that is true but you can still get nice deer anywhere in the state. Truman Reservoir generally has more deer taken than anywhere in the state, it's also a very large area(about 100,000 acres manged between MO dept of conservation and Corps of Engineers.)
I'm assuming you are referring to Mark Twain Nat'l Forest which is in the southern part and not Mark Twain Lake which is up north. The National forest is more of a hilly type country not very big hills compared though, it's also a little thicker. That area also has been where some elk are have been re-introduced so you might even be able to get a chance at seeing one. If archery hunting maybe even checkout reform conservation area, it's archery only. It's where a nuclear power plant is at and the area around it is owned by Ameren UE but they allow our conservation department to manage it for public land use which include hunting via archery only the total area is about 6000 acres. The one nice thing about all of our conservation managed lands is that if it's farm able they will farm it, so you can be hunting public land over corn, beans, etc all being farmed(or contracted out for farming)by our conservation department.
p.s. anytime you buy something in MO you are helping out our conservation department. 1/8 of a cent of sales tax goes directly to them. So thanks!