Yes, oil not water. And preheated. Toss a red hot railroad spike into a gallon of quench to warm it up. I like to use old used canola frying oil. Motor oils are dirtier. It's best if the edge isn't too thin when you heat it for quenching. A thin edge will overheat and be ruined. Blunt it slightly.
Temper it immediately after quenching in your oven. Give it an hour at 450°, then cool and file test. If it's too hard then try again at 475°, then 500°. Above 500° use 10° increments. Give a fresh file a good chance to bite the axe at each temperature. If you can't file it then increase the temperature and temper it again. These multiple tempering cycles are good for the steel, relieving stress.
I think the trouble is both the heat treat and the factory grinding. Sloppy grinding ruins the temper in places. Plus if you have un-homogenized recycled steel (as I mentioned earlier) then you can never get a good edge. But let's assume the steel is at least OK and give re-heat treating a chance.
Assume a simple steel like 4140 or 1050, maybe even 1060.