I am exactly like you. I hate drops (they make me very sick) and refuse to ride ToT and RRC. I do Splash Mountain because it's a classic and "must-do." Also, the drop on that ride is quick enough that it's over before I have enough time to get sick.
With that said, I have done EE once in 2009. Most of the ride is fine. From what I remember, it was very smooth and most of the "drops" were at an angle (like how BTMR curves instead of drops straight down). I feel like these kinds of curved drops don't make me nearly as sick and I enjoy them.
The only part that I was screaming my head off was the backwards part. There is at first, a straight backwards drop, this curves around, but you're in the dark and can't see the track and know how the coaster is moving or how long you'll be going backwards. My advice for that part is to 1) scream a lot because it distracts you from that uncomfortable feeling in your stomach. 2) Plant your feet firmly on the floor of the car because I've read it helps "ground" you and 3) I tried to turn my head around to feel like I was going forwards instead of backwards. This doesn't work too well because you're strapped in, but it's worth a shot.
I've chickened out during the 2 times I've been since 2009, but I'm going again in August and plan to force myself to ride. It didn't kill me the first time and I didn't get sick (did feel shaky for a little while afterwards) and I felt like 75% of the ride was very fun, so I want to try it again!! Also, try finding POV videos on
YouTube so you can get an idea of what to expect.
I tried to be as honest as possible, so hopefully this helps you out!