How to raise grade near the fence without making soil contact

The feasibility depends on a couple of things:

1. If you can remove the existing boards without breaking them. My guess is probably, but you would need to do it carefully. I would use a short 2x6 and a light sledge to tap on the board from the backside.
2. What the spacing is on the boards. If you can just take the boards off one side and put them on the other, that makes things easier. If the spacing is wrong for that, you would need to pull the boards off both sides and then redo them.
3. Whether you are willing to do the work yourself. If you are paying somebody, it may be cheaper to have them tear down the old ones and put up new ones than to carefully remove the old ones and reinstall them.

Having dirt touching the fence will increase the chance that the bottoms of your rails/pickets will rot. This doesn't necessarily mean your whole fence will rot; only the parts exposed to heavy moisture are at risk. Maybe you're OK with having a bit of rot on the hard-to-see part of the fence after a few years; how bad it is really depends on the wood, how wet the climate is where you are, how much sun that area gets, etc.

If you want to protect the fence, I see three options:

1. Cut the bottom end of the pickets so it's no longer touching the dirt. If needed, treat the cut to improve rot resistance (may not be needed on natural redwood cedar, but you said it was pre-treated, so any cut surfaces should be treated in the same way).

2. Add stone/gravel to provide free-draining surroundings as you suggest. (I wouldn't recommend plastic as it's likely to just trap moisture against the fence and accelerate rot.)

3. Remove some dirt near the fence (may be too much work, or against your landscape design goals).