If he is a decent weight then his prognosis is probably pretty good.
On the mouth, are there any sores noticeable? Mouth rot can often occur with poor husbandry and stuck sheds - and makes it painful to eat. A vet would be able to give you something to treat this.
If he tolerates soaking then soaking him to help get the shed off his toes is a good idea, but if he gets really stressed out then its not worth it at this point. If he doesnt appear overly dehydrated you may want to wait awhile before you takle this - as he is alreaady so stressed.
The best thing you can do for this little guy at the moment is make sure his set up is as close to perfect as possible - to help minmize his stress. Im sure you know the basics but just make sure he has three hides (warm, cool, humid) that are nice and snug for him, dish of calcium, dish of water, a substrate such as paper towels, tile, non adhesive shelf liner etc, and most importantly good belly temps (measured with a digital therm) of 88-92 degrees.
Once he is set up like this give him several days to just relax, try not to handle him or stress him out - just let his body adjust to the proper care. Then try offering live feeders and see what he does...if he doesnt actively hunt them down, try holding crickets with tweasers in front of him and walking them around a bit, or dropping mealies in front of him and squeezing them with tweasers so they wiggle. Also, silkworms are a great, yummy, easy to catch feeder to try.
If he doesnt accept live food after awhile you may want to try offering him some slurry. Since you mentioned his weight is ok give him awhile before going to this step, as long as his weight is stable give him a week or more to adjust to his new home and to start eating. It may take awhile since he has been through a lot.
If he is starting to lose weight or isnt eating after an extrended period then i would mix up a slurry (recipe below) and dribble it onto his nose.
As soon as you get a fecal sample from him - have it taken to the vet for analysis, since he has been stressed parasites may be adding to his issues and making him not want to eat.
Getting him checked at the vet is a good idea overall - if there is a medical problem other than just stress they can help treat it (i.e mouth rot, parasites, etc)
I have been doing reptile rescue for awhile and the majority of my cases tend to be leos. In my experience the key is the proper set up and a little time - a lot of these guys are suprisingly tough and will pull through with a little TLC.
"In a blender, combine:
1 small can Hill's A/D pet food (available at most vets)
1/4-1/3 cup Ensure (not chocolate)
1 jar baby food squash
1/4-1/3 cup Pedialite
2 tablets or contents of 2 capsules milk thistle (herbal supplement, liver purifier, found at most drug stores.)
1/2 tsp. each calcium powder and herp vitamin powder
1 large handful of mealworms, added slowly while mixture is blending
Puree all ingredients until completely smooth, and pour into ice cube tray(s) and freeze. Store cubes in zip-lock bag in the freezer until ready to use, then thaw out 1 cube at a time and store in fridge for 2-3 days. I use the empty baby food jar for this purpose, washing it thoroughly after each batch.
Using this mixture, draw a full dropper (eye dropper type) and place a drop on the gecko's nose. It will lick it off, and as it does, slowly squeeze out a little at a time. Most geckos love this food, and will lap it up."
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1.1 Bearded Dragons
2.2 Leopard Geckos
1.0 Uromastyx (Mali)
1.1 Corn snakes
0.1 Mexican Black Kingsnake
1.0 Bairds Ratsnake
1.0 Rosy Boa
1.1 Green Anoles
1.1 House Geckos
0.0.2 Flying Geckos
0.0.1 Red Eye Tree Frog