You didn't say if you have been on it, but if you can handle Splash Mountain, you can probably handle Everest just fine. Yes, EE is a larger drop (but it does curve), but it also is faster after the drop (you circle the mountain twice).
Since you say you don't love high places (don't blame you), when on Everest, don't look to the sides as you go up the main lift hill. Just keep looking forwards, at the mountain.
Everest is a great ride. I have been on several times and loved it each time. I went with some friends recently. One of them likes small roller coasters, but does not like really rough rides (she's OK on Space, Splash & Big Thunder). She enjoyed EE.
Later in the trip, she went on RNR. She did not like this one very much. She said it was rougher than EE, because you do have multiple inversions (which she hates) and I think a corkscrew at the end.
I think you should go on EE at least once. It is amazing to compare how much smoother the ride is to most other coasters that have been built in the past. Then, if you hate it, no problem, you know never to go on it again. You do not go upside down at any point during the ride. Just backwards.
On RNR, if you are unsure about whether you should ride, you might want to surf the internet and find some reviews and spoilers to help you decide. (I always pass on that one, since I don't like inversions, also).