If you do submit a paper copy of your return along with the form 8833, Please make sure that the 1040 first page is marked across the top " duplicate submitted for form 8833: to make sure that it does not go through normal processing again -- that will definitely make a mess.
Often you can submit additional forms needed to be submitted along with the return as follows : fill out a form 8453 and attach the document ( in your case form 8833 ) and mail this in. Even though this particular form is not listed on the form 8453, perhaps it will force human intervention and therefore routing to the proper area for disposition. The first option, suggested by TTLina, with my suggestion is a better option
