I managed to get a good model by creating the faucets as a separate part, and them creating the final model, see attached zip file. Note it is nor properly dimensioned, but show a way forward. Note the force to Nurbs on the link of part 1 is required to avoid problems.

Dear Drew DeVault,

I loaded your model for an analysis and found following:

In the first plane you have angles between lines of 60 degrees.

Try following:
Change the 1. angle to 50° and see what happens.
Then change the angle to 70° and see what happens.
You will see, that the constrains are not as you might expect.

I did not do further analysis, but this might help you to find underconstraints a.s.o.

I found it a good practise to design shapes in an total wrong sice/angle and then bring it to the expected form by defining constrains.

I hope I could help, at least it was the way I was successful.

Best regards
Walter