I have a lot of dives on the Speigel Grove, well over fifty anyway. First off, it's not a wreck: it's an artificial reef. Wrecks entail a lot of screaming, and while there was a ton of cussing as this went down, it was sunk on purpose.
Secondly, I have never had more than a knot's current while diving her and even that was rare. I almost always get to drift down and up. While gloves are allowed, I've never used them. I've had plenty of current on the Duane and the Vendenberg, but never on the Grove. I've always had a nice relaxing dive each and every time.
I agree that your buoyancy and trim should be excellent.
You should have a good grip on your SAC rate and dive planning.
Situational and buddy awareness are imperative as they are for any deep dive.
I often mark the buoy line with whatever I have.
Go slow. The slower you go the longer your air will last and the more you will see.
Take time to view over the sides of the boat.
Most importantly, relax and have fun. Almost every boat here in the Keys will run a granny line from the buoy to the stern on most deep artificial reefs so getting there, even in a current, should be doable. Here's the key: if your hand is on a line, you shouldn't be kicking. It's like climbing a rope, only you're inverted (head down) and it's not nearly as hard.