I would explain that I enjoy having them as tenants, but the situation with the pets makes it impossible to continue the lease.By the way, if they want to stay you should "steel" yourself for the innevitable, "...we'll train them not to bark...."' or "....we'll cage them when we're out and that will solve it....".

There is a dog next door that barks constantly. 

-Colten Bark up a different tree. :)If you currently own real estate, you can use borrowed funds secured by that other real estate for down payment / cash to close / etc, but the payment will still be included in your DTI.

Dogs crap on carpet, throw up on carpet, scratch doors, tear down blinds, and their barking annoys the neighbors.

Our lease states that the pet is approved as long as there is no evidence of dangerous behavior or excessive barking. 

Looks like all bark and no bite.HB 2054, introduced by Del.

Other existing terms of my lease make clear that community nuisances (excessive barking, scaring the neighbors, etc)  on my properties are a terminatable violation. 

Let the dogs bark.

Any problems with the neighbors (loud noises,fighting,barking dogs?

If your house is close to neighbors they will be inconvenienced, the dog will most likely not bite them, but he will bark at them and poop on their lawn; all shepherds are bred to be protective and assertive.