Have you checked the ride height of the suspension overall? Sometimes the front AND the rear suspension sag with age and so it's less noticeable. If the front suspension has sagged, the bump stops will be closer to the axles, and so when you hit bumps, instead of the suspension soaking up the bumps, the axles will just hit the bump stops and transmit vibration directly to the frame and body. The fix for this is either new springs or washers used as spacers under the front coil springs.
You can get aftermarket springs from Moog, but be sure to get the CC868 springs, since the CC870 springs are too soft and reports are they just collapse and are just too thin and weak for the Explorer.
Generally I think adding washers under the front springs are a better option since it costs less and the old springs have already sagged, so the height restoration is pretty much permanent. New springs can sag and cause the same problem all over again, and while the Moogs can ride nicer, they can also be noisier since they have a dual-rate winding, with the top coils closer together rather than the evenly spaced coils of the factory springs.
As far as the body mounts, they can cause body movement if enough of them are in really bad shape, but if they are in good condition and there are no rusted off washers or anything indicating problems, it's unlikely that replacing them will be the fix you're looking for. New polyurethane mounts can still give a slight improvement over old rubber mounts, but it's a bear of a job and depending on how it's done, can be incredibly expensive, especially if the old hardware is destroyed in the process.
Here's a forum thread on the body mount replacement process with photos, showing factory mounts and hardware in good condition:
http://www.explorerforum.com/forums/showthread.php?t=389677
If the body mounts look like any of these, then you probably should consider replacement:
You don't need to measure the body mounts for height or anything like that, they will obviously be compressed due to the weight of the body, but it's more about what shape they are in, as a mount that has been compressed too much will have cracks and obvious visual signs that it needs to be replaced.
You might also consider tires and tire pressure, if you are running cheap tires or LT truck tires at high pressures (40-50 psi) they can contribute to a harsh ride, and on the other end, low tire pressure (under 26 psi) can cause mushy handling and wobbling.
If the radius arm bushings were not replaced when you had the rest of the front suspension done, or if they were replaced with rubber bushings, you might consider getting them replaced again, this time with polyurethane bushings from Energy Suspension. They last longer and are firmer, giving more control and better handling.
Same thing with those sway bar bushings. If they were replaced with cheap rubber ones, they may have broken down or gone soft and so replacing them with Energy Suspension poly bushings will dramatically improve handling.
Some of it may just be from having a solid rear axle, too. There are times when my ride, with Bilsteins and Energy Suspension bushings, still hits bumps the right way and so wobbles the whole vehicle. Nothing you can really do about that except get coil springs and a four link rear suspension, or get a whole new rear axle with independent suspension like the newer Explorers.
. It's hard to judge the condition of the mounts without knowing the new OEM body mounts' height under load though. We don't have harsh winters and much road chemicals so the rubber look "meh"/ok.
