The part "95S-1-12" would work well. Typically, the "left" side of the part will connect up to the Telephone company's demarc wiring or the NID's wiring bus. The "right" side will split off into two paris of wiring. One which is unfiltered for DSL and one which is filtered for POTS. They are self stripping parts, so no need to peel away the wiring jacket when inserting your home's wiring. Ideally, Verizon would replace your NID and install it for you when they replace the NID.
The current method you have now would work fine as well, assuming the filter itself isn't going bad (you would have noise on your phones if there was a chance of it being bad) and you don't have a bad two-way split or any loose splices/terminals. That would leave the problem with your retrains to be outside if your home wiring has been ruled out already. Again, can't stress enough before calling Verizon for trouble - if you have an alarm system, make sure it is behind a filter.