I would find a vet who will biopsy the dermal masses and take X-Rays of your dogs chest and abdomen. The biopsies could be performed either by fine needle aspirate, which just involves a needle, or via punch biopsy, which would involve a local anesthetic, and one or two sutures. The latter method would yield a more definitive diagnosis.
Neither of these biopsy methods involves general anesthesia and are therefore safe in a older dog.
An X-ray would show masses in the chest or abdomen.
If the dermal masses turn out to be benign they would only have to be excised completely if they were creating a mechanical problem for your dog.
If masses were discovered in the chest or abdomen of your dog, than you could elect to have an echocardiogram and an abdominal ultrasound.
Your dog may be coughing because she may have bacterial, viral, or parasitic heart or lung disease. Or, she may have the beginnings of age related heart or lung disease, which could be controlled by medications. Please don't assume she has lung or heart cancer because of the dermal masses. Just have the diagnostics to find out definitively and to set your mind at ease.
I'm not the vet, but just want to tell you it never hurts to get a second opinion. If you aren't comfortable with the advice your're hearing from your current vet, then consult another one. Medicine for animals isn't all that different than medicine for humans, and somethimes you just need another set of eyeballs on a given condition.
I'm a dachshund lover too. My oldest female (nearly 11 now) had 7 of those fatty tumors removed about a month ago. They had to be removed to check for malignancy. One week after they were removed (and they were all benign), we found another one that just suddenly appeared. Took her back to the vet, he did a needle biopsy, and it was cancerous. Needless to say she was back in for surgery the next day.
Find another vet - one who is a surgeon (look for an animal hospital or vet hospital where they do surgeries and have 24-hour emergency care). It's not worth the risk. Sienna's cancer sprung up in one week and had grown so large it was very close to her jugular.
You might also want to do a search on dachshund breeders in your area then call them and see who their vet(s) are. You can Google dachshund breeders or call your local AKC.
Good luck - give your sweet doxies a hug from me and mine.