Rear wheel play

Loosen rear axle nuts.
Pull back on the rear wheel until the chain is tight, then move forward just a bit and tighten both sides with your hands at the same time. Try to keep the wheel centered.
Tighten the right axle nut, center the wheel and then tighten the rear.
Turn the crank slowly. The chain should have slight play at the tightest point in the crank rotation.
If the chain is too tight or too loose loosen the right nut and move the axle forward or back, tighten and retest.
When tension seems OK loosen left nut and center the wheel, then retest.

I’m happy to help further over the phone at https://www.6ya.com/expert/donald_f2ed37026a3ac881

Take the wheel and get it balanced and check to see if wheel is bent. From another place beside the tire people you brought it to.
Differential service or was it rebuilt? Slide in axle or rear end differential may have slack in it. Rear end will make noise if it has a problem.
Could be a bent slide in axle. Jack it up and spin it with wheel on.
If 2wd jack up back and put it in drive.

To remove a 2008 Softail rear wheel FIRST block motorcycle underneath the frame so that theweight of the motorcycle is off of the rear wheel, remove saddlebags if equipped, remove the belt guard and the debris deflector from the rear fork, inspect the wheel bearing end play and service the bearings if and as necessary, label the wheel spacers for left or right locationand orientation i.e. fork side or wheel side, remove the rear brake caliper from the caliper mount and support it using an elastic cord or such,For the FXCW/C modelsthe nut retainer will remain in the axle nut and does not need to be removed and if it is removed it must be replaced. For all but FXCW/C remove e-clip, remove the rear axle nut, and for FXCW/C remove the rear axle nut, loosen both axle adjuster screws an equal number of turns to remove tension from drive belt, with a suitable drift or other tool tap the axle towards the right side and remove it, remove the spacers and the caliper mounting bracket, move the wheel forward and slip the belt off of the sprocket, raise the motorcycle to allow enough clearance for removal of the rear wheel and pull the wheel with the belt sprocket from the rear fork.

To install a 2008 Softail rear wheel FIRST apply a light coat of wheel bearing grease to the axle, to the bearing bores and to the bore of the inner sleeve, roll the wheel into place in the rear fork and slide the drive belt over the drive sprocket, position the left bearing spacer between the wheel and the fork, from the right side carefully insert the axle (with its washer for FXCW/C models) through the right rear fork, the short spacer, the rear caliper mounting bracket, the long spacer and into the bearing bore, continue sliding the axle through the wheel hub sleeve, the left side spacer and the left rear of the fork, install the washer, the axle nut, and the retainer, verify correct axle alignment and the check belt deflection, check the wheel bearing end play after tightening the axle nut to the specified torque because excessive end play can adversely affect stability and handling and insufficient end play can cause bearing seizure, tighten the axle nut to 95-105 ft-lbs (128.8-142.5 Nm), install the belt guard and the debris deflector, install the brake caliper and pads, install the saddlebags, if equipped, pump the brake pedal to move the pistons out until they contact both brake pads, verify the piston location against pads, verify the position of the brake pads against the brake disc, rotate the wheel to check for any interference(s) or misalignment(s) and test the brakes for proper operation.

wheel bearing is the first to be check. lift the car on the lift and push/pull the wheels in and out. if excessive play replace the bearings.

Check air pressure in tires.

block front wheels and jack up rear slightly off the ground, keeping your body parts out of harms way you can grab and shake tires, if they move back and forth or in and out even slightly it is time for bearing work

It sounds like one of your rear wheel bearing hub has gone bad and needs to be replaced before you lose your wheel. I am thinking to pay attention to the right rear wheel hub bearing than needs to be replaced, they go for around $170.00 at Auto Zone and they can give you the step by step instructions to get the job done with common tools or you can use there loner program if you need to borrow a tool you may not have. With the car parked walk to the rear wheels and push and pull on the top of the tire (Side to side), feel and listen for any play. You should not hear or feel any play at all unless the wheel hub has gone bad. It take less then 1-1/2 hour to change the wheel hub and less if your good with tools. Good luck and keep me posted friend. Be safe.

Bearings? If the wheel is jacked of the ground, can it be shaken back and for to determine any "play" or looseness? If so, hub/bearings are bad.

Hi

Sounds more a rear wheel bearing problem ?

Check for excessive play, and noise from both rear wheel bearings and hubs, with the vehicle jacked up and supported safely.

Good Luck !

A FixYa rating would be nice...Thanks... John.

At that mileage it could be, and most likely is given your description of the noise. I assume it reduces in volume as you slow down. If you can find a quite piece of road drive up to where the noise is apparent and lightly swerve from left to right. (Don't roll the car!! ) If the noise reduces as you swerve left then its the left bearing. If its when you swerve right that the noise reduces its the right bearing. If its both the only way is to jack the rear up and spin the wheels as fast as you can and listen. If there's a hint of a rumble from it, its that bearing or both. It should be completely quite apart from the slight hiss as the shoes rub on the drum.
35K may not seem much but if you don't know the history of the car it could have hit a bad pot hole or even had a knock on a kerb.
I have replaced bearing with a lot less miles than yours.

Cheers John