Dead limbs could be removed at any time. minor pruning can be done as needed as well. Major pruning within the canopy should be done while dormant as you note. Late winter is better than early winter to avoid cuts being exposed to excessive cold. Your tree has gone without pruing for some time, so your recovery efforts should not be done in a single season to avoid stressing the tree. We have also gone through 2-3 years now of stressful weather. Be sure you feed and water the tree to aid in recovery. Consulting a certified Arborist(s) for several opinions on treatment can be very helpful. Check out
http://illinoisarborist.org/ for an arborist in your area.
