thank you for your very helpful reply
Shelters are full of dogs with problems. It is really sad that prior owners are not honest about the reasons that have caused them to surrender pets to shelters.
It sounds as if you have a double problem with your new pet. A gastrointestinal problem and a behavioral problems. If one is eliminated, hopefully the second can be addressed without too much stress.
Dog feces doesn't smell good of course, but it should not be foul. Plus, she has an aberrant eating pattern, which may be caused by a gastrointestinal problem.
Gastrointestinal problems can be caused by anything. The list of potential gastrointestinal problems is extensive, and parasites are very common causes of GI problems.
There are many GI parasites, but some common parasites include: coccidia, giardia, roundworms, whipworm, hookworm, tapeworm, and many more.
A fecal analysis may show parasites, but some parasites are intermittently shed and are hard to detect. Please take a fecal sample to your vet for analysis, and, even if the fecal analysis is negative, it might be best to have her treated with a broad spectrum anti-parasitic medication, such as Drontal Plus, Panacur, or equivalent. You should also ask your vet to place your dog on a probiotic, such as Prostora.
If these are ineffective, your dog should be assessed by a vet for other gastrointestinal problems, such as IBD, Pancreatic exocrine insufficiency, and many more. Endocrine disorders can also affect the GI tract and behavior. Your vet can test for these and other possible problems also.
The behavioral problems do sound as if they will take a bit of hard work on your part and may even require professional training and/or behavioral modification medications, which are available from your vet.
Crating would be a good idea. The crate should be small enough so that she cannot escape feces if she does go in her crate.