Had to replace one of my bearings last week. I tightened the axle nut as tight as I could with the channel locks which was more than enough to get the cotter pin through. I would estimate I have around 1/8" to possibly a 1/4" of play when I jack the tire of the ground and give it a tug. I know you don't want them too tight but this seems excessive. How could I get it any tighter or should I just take an impact to the axle nut?

I normally tighten them down hand tight w/channel locks, good and tight to seat the bearings and then back off just to the point where you can insert the pin. This should allow enough play to spin freely and not over heat. Add plenty of red bearing grease and you should be good to go. I wouldn't use an impact...too much force.

You should not have that much play, none at all really. Sounds like one of the races was not in all they way and it moved. Usually I tighten down the nut till no play, then lossen a little to get cotter key in.

quarter turn counter clockwise after tight max.

Tighten all the way by hand....no play. Back off a quarter turn. There will be slight play but not much. TO tight is just as bad as TO loose. Often when bearings go bad the race, spindle, hub or drum also are bad even if they seem ok.