You can usually stretch a Shimano derailer rating by at least a couple of inches, so a 34 should work with the existing Sora derailer.
I rode from Glacier west on the NT. Most of WA 20 is limited to 6% grades, which is doable with 20 gear inches. IIRC, there were a few steeper stretches on Loup Loup (which I didn't mind, as I was going downhill!), but they were fairly short. Also, there's 3/4 mile of 8% going east of Tonasket that was aggravating because there was a school zone and stop sign right at the bottom -- nowhere to let it run out. Grr. All that climbing gone to waste.
But I digress. You'll have to climb that, which will be easier early in the morning when you're fresh. The rest of the Washington passes, and Idaho and western Montana, was not a major problem. Though we had some 3,500 miles in our legs by the time we got to Glacier, which might have some bearing on my perception! Also, if the NT is like the TransAm, the worst grades are in the east (Virginia, Kentucky, and Missouri on the TA).
Finally, don't be ashamed if you need to walk a few hundred yards. It's still human powered travel. And while road builders can build roads you may not be able to ride up, even when they truck the equipment and material around to the top so they can pave going down they can't pave a road you can't walk up!