I would jack the machine up and run it. Does the it do the same thing? Something is going on with one or both drive motors causing excessive load on the engine. There is nothing wrong with the engine as far as I'm concerned. The engined doesn't know which way you are traveling, it merely reacts to load. If it is pulling down that bad only when backing up, that points to a drive issuse.I'm not convinced that the issue is the filter, the water needed draining, that's for sure.
Only bogging down in one direction is very strange.... A stuck valve would just give a lack of drive power, not bog the engine down like that.
Black smoke is too much fuel, not enough air. That can be due to excessive load or a restricked air filter, so check the air filter too.