What Guya said.
They are cutting staffing at the IRS dramatically, so anything that requires human intervention (like a paper return or one that requires a human to review it) may take exponentially longer to process. So doing anything you can do electronically is probably going to be the speediest. If your tax prep software service allows you to complete the 8833 online, you'll be ahead of the game.
When I filed from the UK I sent paper copies. I believe I wrote (in very small writing) on the line "per UK/US Tax treaty" and referencing the section of the treaty, next to where I entered the sum of $0 as being taxable. (Sorry, those papers are in storage or I would look for you.) It worked just fine.