You can do better than 44mag for a similar price. As mentioned the 45/70 and 444 Marlin might be better choices. Probably 450 Marlin also if this list has been updated since 2014
These specific straight-walled cartridge rifles are legal for Ohio deer hunting: .357 Magnum, .357 Maximum, .38 Special, .375 Super Magnum, .375 Winchester, .38-55, .41 Long Colt, .41 Magnum, .44 Special, .44 Magnum, .444 Marlin, .45 ACP, .45 Colt, .45 Long Colt, .45 Winchester Magnum, .45 Smith & Wesson, .454 Casull, .460 Smith & Wesson, .45-70, .45-90, .45-110, .475 Linebaugh, .50-70, .50-90, .50-100, .50-110 and .500 Smith & Wesson. Shotguns and straight-walled cartridge rifles may have no more than three shells in the magazine and chamber combined while deer hunting.
The current reg in the 2018 book just says
Straight-walled cartridge rifles: All straight-walled cartridge calibers from a minimum of .357 to a maximum of .50.
So a 450 Marlin should also be legal.