What would cause the front suspension of my 4x4 to feel like it is riding on the edges of the tires going around a curve and constantly moving steering wheel from side to side?

Sounds like a wheel bearing is going out. Put your Explorer on a floor jack and raise the left front tire off the ground. Grab ahold of the tire and see if you can make it move any by pushing/pulling on the top side or bottom. Any play (movement) in the tire/wheel indicates a bearing might be out or on its way out. Is it a 4x4 or a 2X4?

Front end suspension parts faulty

Possible broken belt in one of your tires. Move the 2 front tires to the rear, see if the problem follows the tires. Or run your hand around the circumference of the tires to feel for a bulge. Other, more expensive problems could be in the suspension. A-arm bushings, or wheel bearings.

I'm guessing the lower ball joints are shot. This will cause the tires to lean in on the top and and wear the inside of the tire. I could be wrong! Try jacking the front wheels up and support the vehicle on jack stands, now grab the tire top and bottom and try to rock it in and out. If there is movement, you will have to have someone watch to see what is moving, while you rock it. If it's the ball joints, they will need to be replaced before you can align it. This can be spendy, there's a lot of labor involved.

track rod end maybe

I will try to help you locate the problem.First check the steering and suspension:
NOTE: Park yhe wehicle on level ground, turn the engine off and set the parking brake.
1. Visualy inspect the steering and suspension components for damage and distortion, Look for damaged seals, boots and bushings and leaks of any kind.
Clean the lower end of the steering knuckle. Have an assistant grasp the lower edge of the tire and move the wheel in-and-out while you look for movement at the steering knuckle to control arm balljoint. If there is any movement the suspension balljoints must be replaced.
Grasp each front tire at the front and rear edges, push in at the front, pull out at the rear and feel for play in the steerin system components if any freeplay is noted, check the idler arm and the tie-rod ends for looseness.
Please let me know if this is of any help.

Tires need balanced or you need an alignment. Make sure tires are inflated to max cold psi

Fit softer shock absorbers if you want a softer ride, or if it rolls around too much, have a tortion bar fitted.
Changing springs to a performance type will also affect things, but talk to the manufacturers before selection.
Fitting different tyres will also affect handling and ride.

could be a bd axle or ball joints wheel bearings are also a concideration

What are the tire pressure and how many miles on your shocks??
The maximum tire pressure marked on the tire sidewall refers to the pressure required to carry the maximum load of the tire, and is generally not the same as the manufacturer’s recommended tire pressure for your vehicle. To find the recommended tire pressure for your vehicle’s tires, refer to the information label, which is usually located on the edge of the driver’s door, or the door post or other conspicuous location. If you can’t find the information label, check your owner’s manual.
An overinflated tire rides on just the center portion of the tread. The smaller contact area means reduced grip on the road, leading to a harsh ride, handling issues (such as steering and stopping problems) and increased wear on tires and suspension components.
   The shock's also have an average life of around 50,000 to 80,000 miles and should be replaced by then. 
Good luck and hope this helps. Start with checking the tire pressure, keep me posted, ne glad to help friend.