Poor little huge Tank. Bassets are pack dogs, and like to 'belong' to people/with other dogs - she's probably unsettled and is a bit anxious about being left again. Hartley was 3 when I rehomed him, and he did cry for a few nights when I made him sleep in the kitchen on his own, but he soon settled in. He never, ever howls (and I've recorded him while I'm out to make certain!) I'm sure once she's found her feet in your set up, she won't be quite so clingy. Unfortunately, they do have quite loud barks - designed to be heard over fields - but they're not a barky breed, generally. You might have to do the 'go out for 5 seconds/come back before she starts barking. Go out for 10 seconds, come back before she starts barking' thing, just to teach her you're not leaving. Was that one of the reasons they were keen to rehome her, by any chance?!
It's good that her teeth are OK - that can be expensive - and it's best to get some weight off her, to help with her joints. Ignore the begging sadface as much as it's humanly possible. Try her on low protein working dog food, like Skinners Salmon/Rice - my dog scoffs it, and it's relatively cheap. Has she had much outdoor exercise? Bassets might look like couch potatoes, but they love a walk, and should be able to cover a fair few miles quite happily.
Eyes and ear issues are a common Basset thing - ears aren't too hard to keep on top of, but you need to be quite diligent about it, as hounds are pretty stoic and won't tell you there's a problem till it's very bad. They're like cocker ears, basically, but bigger. A quick wipe round the outside with a baby wipe a couple of times a week is usually enough to keep the basic gunk in check; you can get ear cleaner off the internet for about £8, and a squirt of that, rubbed in for a minute or so (get the vet to show you how to massage it down the ear canal) keeps the bacterial levels right.
Eyes can sometimes look worse than they really are. Was the '3rd eyelid protrusion' cherry eye? It's not uncommon and can be fixed with a straightforward op. Hopefully - if that's the right way of putting it 
- the dry eye hasn't been treated, and your vet can sort it out once and for all. Again, maybe worth emailing the breed club? I had some really excellent long-term-owner reassurance about medical issues that one vet had really put the frighteners on me about, and turned out to be fine in the end.
Sorry for the essay, this is probably very biased because you sound like a thoughtful, caring dog owner and I want Tank to have a loving new home! Of all the breeds, Bassets have the biggest hearts and so much gentle devotion to give - when she's curled up on your lap, snoring happily, you'll be amazed at how small she can make herself seem...