Shakes when you accelerate above 52 mph--feels like the shaking when rotors need turning. Had tires balanced--still shakes.

Do you feel the shaking in the steering wheel as well? If so, it may be a worn steering linkage. If not, have your CV joints and U-Joints checked and, if you have a 4WD, have the transfer case output shaft bearing checked.

Your tires are either need balancing or you have a broken belt in your tire

Try balancing tires.

OK, when the ABS module is unplugged, the same results. That tells me that the ABS system is not working properly.
Did they look into the master cylinder and other abs related components?
Any ABS codes on it?

If steering wheel shakes above 50 mph that's likely to be the tires on it, rather than a brake problem.

Have tires balanced and rotated and inspected for out of roundness that might cause vibration above 50 mph.

there are a few things that are more likely than others .. first tire balance .. if a tire loses a balancing weight or gets its rim dented or if the tire is simply worn .. then you need to rebalance the wheels .. this should be done every year or so anyhow .. a broken cord or belt in a tire will also show up when they try to balance as will a bent rim .. this will tend to shake the car above 50 mph .. thats also not a safe condition since the tire could fail unexpectedly .. you might observe a tires tread wiggling when observed moving from another car or on a rack when you spin the tire .... tires are most likely the trouble but a worn out CV joint will also shake at certain speeds .. usually this shows up at low speeds or as excessive slack in the drive system .. warped rotors are a frequent source of vibration . but usually the problem is most noticable as brakes are applied .. the best bet is to have the tires balanced and inspected .. that will give you lots of information .. one more thing you dont see as much anymore is wheel vibration at higher speeds caused by worn ball joints or tie rod ends .. the wheels will actually oscillate above a critical speed . you will often see cupping on the front tire treads if this happens .. that cupping problem can also be observed when the tires are balanced .. the tie rod ends can be checked with the wheel off the ground .. shake the tire left to right .. excessive slack would mean worn tie rod ends ...

It sounds like you may have more than one problem here. First I would investigate Front End Alignment and ALL Steering/Suspension components for excessive wear and/or looseness. Second I would Inspect the Brake Rotors and Pads for uneven wear. Third I would check the Tires for uneven wear and balance. Many things can cause these symptoms: Worn out Ball Joints, defective Shock Absorbers/Struts. Worn out Tie Rod Ends, Idler Arm, even the Steering Gear Box can be part of the problem. Brake Rotors and Tires can be out of Balance with either one causing the problem over 60 MPH. I strongly suggest having your vehicle inspected by an ASE Certified Front End and Brake specialist immediately. It is very dangerous to drive a vehicle with these symptoms.

Most likely with this problem is the front tires. The tires maybe " out of balance " or " out of round " causing the vibration at certain speeds.

Rotors will only cause a shake when braking. Your problem is more than likely out of balance tires.

The best brake components are usually the ones that cost the most.

ball joint
control arm
shocks
brake rotor
and all your suspension parts have them check by your mechanic

Get your tires checked out, you may have a seperated tire, or may need to get the balanced checked. and check your air pressure. then while your at the shop get your suspension checked out. while its up in the air.