brake problem

On my 69 GTX, under standard braking conditions, when I apply my brakes (both EASILY or hard and it doesn't matter on the speed), my right rear brake will lock up (even when applying minimal pressure). It does let go if I remove my foot from the pedal, but will lock up again if I so much as touch the pedal. I have an idea as to the problem, but would appreciate an unbiased response from you as to the issue, AND the repair. FYI, the brake system was gone through just 6000 miles ago and the rear shoes look good (and even) on both sides.

Poor adjustment, possible sticking cylinder, grooved backing plates

When adjusting shoe clearance, have the parking brake cables slack

Sounds like the hose is rotted inside, but confusing you only have trouble with one wheel. Could be adjusted closer back there and a slight drag shows up on that side and not the other.

The other listed ideas are great ideas too.

Backing plates seem fine. I don't see the shoes hanging up on either side on either wheel. The hoses look fine too on both wheels. Everything was put on brand new and ALL components have only abut 6000 mile on them.

So when you say "all components" I assume you mean the springs and pins were replaced as well (which is what I was going to recommend).
The next thing (IMO) is to remove the drums (both) and inspect everything, ensure you don't have the long shoe on the front, springs are where they're supposed to be, shoe pads are clean/dry, etc.
Theoretically you have the same hydraulic pressure to both sides. So that excluded, it has to be something mechanical.

Yes,all components are new. And I did check to make sure the shoes were on correct. Just removed drums and everything is clean and dry. No axle grease leaks. Hit the entire assembly with brake cleaner. Backed off the star adjuster and reset to proper specs. Re-tensioned E-brake cable. Brake fluid in master is at level. Just took car for test drive and a strange thing happened. When I eased in to the brake, the right rear did not lock up, BUT the brake light on the dash came on as I pushed the peddle harder. Did not continue the test drive. Sometimes it takes 10+ minutes of driving before the right rear locks up, but I don't take chances when it comes to brakes. So I don't know if the lock up is fixed, OR, I have a new problem.

Curious on what brake fluid are you using?

Wrong self adjuster on that side?

That's a good guess, or installed wrong.
is the ebrake cable possibly sticking on that side?

Your warning light coming on is either Ebrake or pressure differential valve related, Could also be a pinched wire but doubtful. Disconnecting the two wires will identify which is causing the light to come on.
Do keep us posted

I believe you have two separate issues.
The last issue is a different issue then the previous issue.

For the 1st issue, "Backed off the star adjuster and reset to proper specs." Did you do this on both sides, or just the right side? If you just did the right side, you may have the star wheel too loose.
How did you determine "the proper specs", and how did you set it to the specs?

"Re-tensioner the E brake cable." I assume was done with the brake drums in place, how far out of adjustment was the E brake?

"Brake fluid in the master is at level." Did you add fluid?

Issue #2. The brake light on the dash that comes on during driving is turned on by the proportioning valve in the brake system. The proportioning valve piston being pulled off center to one side or the other, which turns on the light. It happened when one side doesn't have enough pressure, and the side with the pressure pushed the valve off center. This usually happens when there is a sudden fluid loss like a brake line failure, or possible the shoes (or disc brake pads) don't have enough travel to seat against the drum (or rotor). You need to look under your car to make sure you do not have a fluid leak. that leak can be anywhere from the master to each wheel cylinder (or caliper). If no leaks are present, you need to look for the reason the shoes can't contact the drum.

Since you didn't have this problem until after you made the brake star wheel adjustment, I believe you have the star wheel(s) set too loose and the shoes are not setting against the drums like they should before the pedal ran out of stroke. applying more pressure when the front brakes have applied but the rear have not will push the proportioning valve off center towards the rear brakes that it senses has no pressure. It falls in line with "the right rear did not lock up, but the brake light on the dash came on when I pushed the pedal harder."

I did add the e brake adjustment because if your adjustment procedure happened to pull the e brake pedal off its resting place, that too will cause a brake light to light. Some models have the e brake "om" light in a different location, but some use the same light as the proportioning valve uses.

You may have sort of fixed one problem, but may have created the other.

If you back out of a driveway and turn the same direction every time, its possible for one rear brake to "self adjust" tighter then the other. This often shows up after a few thousand miles after a full brake job. The repair is to back off the star wheels on both sides and reset the adjustment. You may have fixed that problem.
If when you adjusted the E brake and that adjustment pulled on the brae pedal, it may not take much to pull the e brake off its resting point and turn on the e brake light. Applying more pressure to the brake pedal trying to get the right wheel to lock up could have moved the levers in the drum enough to pull the e brake off its resting place and light that light.

Start by looking for a fluid leak 1st.
If no leak is found, the next step is checking to make sure the E brake pedal is completely released and against the switch that turns on the light. Releases the e brake and physically pull the pedal up and see if it stays up or wants to pull back down. see if there is still a brake light lit on the dash.
If all that is good, pull both rear drums and back off both star wheels, make sure the tops of both shoes are against the center anchor bolt, then replace the drums and reset the brake drag adjustment with the drums in place.. You may have to bleed the front brakes to reset the proportioning valve.

Give the brake pedal a good hard push. that sometimes resets the valve and puts out the light.