Your care and consideration of little Rusty dog is heart-warming to me. You are very kind.
Obsessive licking of the skin and random surfaces can be a sign of skin disease, nausea or possibly anxiety (however be aware that nauseous dogs also appear anxious so it can be tricky to tell from observation). I would certainly encourage you and your partner to take Rusty to your local vet and have a discussion with them about all the signs that you notice in Rusty. Depending on the physical exam and your feedback, the vet may choose to run some tests, for example, skin tests, bloods and/or abdominal imaging, in order to potentially find the cause of the compulsive licking.
If they have ruled out other medical causes, then it is possible that this behaviour related to anxiety. If this is the case, speaking to your vet about how you can manage the anxiety with medical and/or environmental adjustments would be useful. Also be aware that veterinary behaviourists are out there and are brilliant at managing many conditions including different forms of anxiety.
I really hope you and your partner can get to the bottom of why darling Rusty wants to lick obsessively. Hopefully then you will find a resolution as well!
Keep us posted with how Rusty goes! All the best!