Just replaced axle on rear wheel. It is loose at first but as soon as I ride it, it tightens and wheel no longer rotates. Help!

Perhaps I have put in too many/too few ball bearings? ... Should they sit snug, some slightly above the others, or all on same level with slight gaps?

All the balls have to touch both the cup and cone surfaces and therefore should not overlap. If the balls are overlapped they will bind up as the axle rotates.

Did you use a guide to rebuild the hub? Have a look at this Park Tool Repair Help article and this RJ Bike Guy video.

I have sometimes made the mistake of not tightening the locknuts enough, causing the rotation of bearings to drag the cone, tightening it enough for the bearing to bind.

The solution is to properly tighten the locknuts.

I'v been also in the situation of not having the right tools to remove the freewheel, which makes it more difficult.

What I've done to correct the situation is to properly locate the freewheel side cone on the axle, and then tighten it with its spacer and locknut when out of the hub. For this stage the axle and nuts must be free of grease and oil so they are more likely to keep the setting.

With this sub assembly done, put grease on the bearing races and put the balls. Grease is thick enough to hold them temporarily (I do not use the bearing cage in order to fit as many balls as possible. Also, some freewheels have a center hole so small that a cage can not be installed through it.)

Insert the axle from the freewheel side and thread the oposite cone. Adjust the preload and tighten the locknut of that side.

This last step can take a bit of trial and error for the once-in-a-while home mechanic.

My criteria for setting the preload is: Axle does not move side to side if trying to wobble it with my hand. If I try to spin the axle it is doable but won't spin freely (due to grease viscosity). Upon installing the wheel on the bike, it must not move sideways but should spin freely.

The worst case I had was actually a hub that was too worn and had damaged races, so at one point during usage, a ball broke and bind with the cone, dragging and tightening so much that it bent the center tube of the hub. (cheap steel hub).

Bottom line is check hub races, cones and balls for damage. Balls and cones are cheap, if you find surface damage or cracks or any deformation, change them. Some hubs, even some cheap ones have replaceable races. So, if possible and economically feasible, change then when damaged.