Is there any way to do Season Unending if the Civil War never started without trading The Reach for The Rift?

I have things to do in those cities and didn't want to change jarls yet.

Short answer: no.

Long answer: you can try to convince your faction leader to accept a minor hold rather than swapping the Reach and the Rift, but the opposing faction leader always gets what he wants.

So, if you're Imperial or unaligned, Ulfric will get the Reach, and if you're Stormcloak, Tullius will get the Rift.

So go do those things first. Everything waits for the Dragonborn.