If you're actually going to walk home, you're free to use either "Jeg går hjem" or "Jeg drar hjem".
If you're leaving for home by any other means of transport, then "Jeg drar hjem" is preferred, but "Jeg går hjem" is still passable if you still have a short walk out the door before getting in your car for example. If you're standing by your car while saying it, then you're going to sound a little silly.
å gå = to walk
å dra = to go/leave
If you're lacking context, then always default to "å dra", as using "å gå" is specifically related to walking while "å dra" specifies no mode of transport.
Just a comment to hopefully add to the pageant: in Englesk one can say 'draws away from' or 'draws closer to' which must come from the same root. There is also the sense of pulling since technically wire, for example, is made by 'drawing' or pulling malleable metal through a carefully designed hole.
Same as i read a comment that helps with hit and dit, think of them as hither and dither (archaic english for here and there). I revel in the irony that hither and dither may have roots in old norse he(insert 'th' symbol, my phone keybord sucks. Looks like d but has line through the vertical stem) er i think.
I'm not a native speaker so I might be wrong,but I think you can't say that because the verb "å dra" implies motion, therefor you should use "dit". "Der" is used when you're stating a location (no motion). For example:
Hun drar dit - she goes there (motion) Hun er der - she is there (location/no motion)
Same applies to hit/her:
Hun drar hit (motion) Hun er her (location/no motion)
"Dit" indicates movement to a place, but away from the speaker, like the old-fashioned "thither" in English (if it was the opposite direction, towards the speaker, it would be "hit"/"hither"). If she was leaving "from there", then it would have to be "hun drar derfra", I think.