Minimizing sway is normally a combination of different things. Good call on upgrading to LRD tires. I didn't see where you stated what the TV tires are. You want LT tires and raise the psi in them. I run our LRE at 80 psi at rear and 75 on front for towing. Shocks on a TT can really help a lot and sure did on our TT. Not a simple bolt-on project tho. HD Bilsteins (or equivalent) on the TV can also help.
FWIW, if going through BC to Alaska, gas prices are predicted to be up around $8-9 USD/gal due to a war between BC & Alberta gov'ts. Alberta is threatening to cut off oil supplies to BC unless it stops it's objection to the Keystone XL project which is the heart of the issue. Gas is already over $7 USD/gal here and it always goes up a lot over the summer. You might want to stay out of BC for as much of the trip as you can. I just filled up in Bellingham WA at $3.13 USD/gal.
