Cat with some sort of digestive / intestinal problem?

Ok, have had this cat to the vet repeatedly and no real relief. We've had many cats over my life, never seen anything like this. What could it be?

Symptoms:

Smell.

She had burps that smell like something died in there.

Her bowel movements have incredible stench and it is not a poop smell. It is a something died smell.

When she coughs up hairballs, they also have that stench.

Pain.

She seems to have pain when picked up like you'd normally pick up a cat under their stomach.

She screams like it is a sudden very sharp pain when coughing up hairballs.

She has constant diarrhea.

We have done everything according to vet instruction. She's been treated twice for parasites, once being kept boarded to make sure it was treated, and twice for an unknown bacterial infection. Once treated for both at same time. Also had her teeth cleaned. Also has been on first cat food with no wheat and now a special cat food they usually only give right after surgery. She seems to have slightly less diarrhea with the after surgery food. After they treated both together, for a brief period the stench went away and the diarrhea seemed to almost go away. We took her back when it started coming back and the vet said negative stool test so nothing was done. Any ideas? Suggestions? Is there maybe a vet specialist we should look for?