Left rear wheel is locked up

Sounds like your emergency brake is either hanging up or there is a interference issue with your new tires.

I would start with making sure your emergency brake is releasing completely.

First thing is to find a level place to raise the left rear wheel. Place a block behind the right rear wheel.
Place the transmission in neutral (disengage 4 wheel drive if you have it.) release the E brake and spin the wheel to check for drag. If your wheel is hard to turn and or you hear or feel some resistance the E brake cable may be binding, the shoes may be hung up, out of adjustment or worn out.

AWD vehicles have a centre diff in the transfer case and if there is no traction on the rear wheels then the front wheels will not turn. All wheel drive is not equal to 4 wd which you assume apparently. Check that there is not a diff lock button in the car to lock the centre diff turning it into 4wd vehicle in which case the front wheels would have moved the vehicle

Could be your e-brake cable connected to your back shoe.

Check the transfer case. Put a jack under the rear axle and jack up the rear wheels. Have a helper put in 4 wheel and the trans in drive. Have him/her ease off the brake and see if the rear wheels spin. If they spin, 4x4 isn't working. If the truck tries to pull forward with the rear wheels up then it works. Now, if the rear wheels are spinning carefully look under the truck and see if the front driveshaft is spinning along with the rear driveshaft. If the front shaft is spinning then it is an axle problem. If the front shaft is not spinning then it is a transfer case problem. GM has a problem with the NP263 case.

you have a 580 ck backhoe and a red box so I am not as to what the red box is referring to . As far as the case backhoe I believe the 580 ck is a hydromagnit and is a 1985 to 88 vintage there are two linkages on one side to do quick turns you can stop a wheel and turn that direction while turning the steering wheel to the same direction but this is a brake adjustment the two pedals on lets say passenger side for clarity or left side looking from the from has two pedals the one on the far right is the pedal for right side and the one on the left side is for the left brake, on the oppsite side of the machine you have a foot pedal that disengages the transmission by pushing down on the pedal and there are adjustments to that pedal as well you can shorten the Travel of the pedal by loosening the locking nut and then turning the clevis's after you have removed the cotter pin and the clevis pin then you can shorten or lengthen it's travel to change it. From off the top of my head I believe the most common problem with these backhoes was the clevis pin rod would get bent or the clevis pin would wear out and cause slipping from the transmission, but it is not a clutch per Se it is an actuating rod that engages the tork converter There is also a valve on the back of the transmission that is electrically engaged that comes loose that would cause it to slip. so after checking that the clevis pin rod is straight then check that the differential has oil in it and also the transmission has oil in it if either are low in oil it can cause it to slip so check that too. there is the possibility that the forward and reverse handle may be out of adjustment and or worn causing it not to work but you will have to go step by step, check fluid levels also have you changed the transmission oil filter? this can cause it to slip as well. good luck finding out your problem check the top transmission oil level too if you have not done so as this could cause you problems as well. if you need any more information just post to this answer and I will try to help you as much as I can but your better off buying a book on it to figure your problem out if your don't know anything about this tractor.

check the connection of your control circuit

sometimes it is necessary to put in reverse and then forward after settting 4 wheel. That ensures all pressure on any gears is removed. The older the vehicle the more the gears need help to align and engage. Good luck-Ned

I'll assume this is a manual 5peed. The transmission (a W5M33) has a viscous coupler (VC) inside that is linked to the transfer case shaft. This in turn connects to the prop shaft to the rear diff. The rear diff (for the TSi) is usually also viscous coupler to provide a limited slip (LS) function. So ... what can cause the rear wheel lock up? If the W5M33 VC breaks (unusual but not impossible), you'd probably still have rear wheel spin. If the transfer case brakes (quite common with hard use), it could lock up, locking the rear wheels. The W5M33 VC then would overheat. A VC is filled with silicon that smells vile when heated. Best case, you'll have to pull the transfer case off and inspect it. Worst case, pull the whole transmission out. If all is OK with the wheels off the ground and a rear LS, spin one rear wheel and the other does the same. On the front, no LS function, and one wheel will spin. Look at lancerproject.ca for more info on AWD transmissions.

Hi First of all check the adjustment on the rear wheel thats sticking sometimes the brake pad comes away from the metal it is fixed to this maybe the case on your vehicle, the make & year of vehicle would help further but I a sure you got something to check now. Please leave me feedback, Malcolm Campbell.

okay you had brake work done!!! "THINKING" that was the problem originally right rear wheel locking up! okay,,, so doing brakes has not solved your problem! I would maybe look into rear end bearing??? do you have any load road noise from rear and do you have rear end fluid in differential? question to you was the baking plate ""WET"" looked damp? or moister get at it!??? seal might have leaked all fluid out??? all the above would be good place to start! good luck I hope this info helps you out.Happy new year.