Can they put up any longer video's of her limping? Walk her in a straight line if possible. I see her taking a short step in the rear and also swinging that leg out ever so slightly. Yes she stretches out the front legs but I think that was just to warm up getting up and moving. Personally if they have not done x-rays thats what I would be doing is, hips/knees/elbows/shoulders. Bone spurs or OCD could be also coming into play.

I have found that it's typically most noticeable when dogs have ailments when they first get and start moving around or after a busy day. The middle part of the day you will barely notice it because the animal pushes threw the pain.

With the dog carrying extra weight this would be difficult but have they checked for any muscle loss on her? Any leg they are favoring over time will lose muscle tone.

Seriously weight management is Most important with a dog like this and if she was mine I would put her on a diet and 2000 mg of fish oil.