any evidence of worms in her stool? She is doing that motion with her beak like she is trying to move food from her crop.
i had a hen with worms once who had some in her crop and was doing the same motion "burping", also lethargic. we had to deworm the hens and she got better. t
his same hen also has had sour crop.
examine her crop, is it hard, or feel like it has foreign objects in it- not just food and small pebbles. anything large or even moving might be worms
does it feel like a full balloon, soft and really balloon like? does the air smell foul coming from her mouth or when she does that "burp" movement. if you squish gently on her crop does air come out/ smell bad?
please bring her inside and begin treatment after assessment. she will need to be fed foods high in protein such as cooked egg, greek yogurt, and water with electrolytes from a dropper