Any suggestions on a product to push through my pipes and clean out the gunk?

Im sure speedball1 will have THE answer to this one.

Not sure about putting drain cleaner into the disposal. I imagine it might help clean it out, but it might also corrode and shorten the life of the unit. You'll need to hear from someone who knows better about that.

Is is just the kitchen sink that is slow to drain?

It is not slow to drain, just can be smelly... We have had the house for 2 years now and I think the previous owners were not the cleanest around...