I think you're spot on when you try to play Rek'Sai like Shyvana. She has fast clears and can run around the map really fast. I think Rek'sai has a bit better counter-jungling and ganks though because of your tunnels and knockup. You can use this to sneakily tunnel in behind enemies if they are over extended (especially top or bot) so you can get some really sexy ganks off. Also, you ult is great for map-pressure. Try showing top to draw the enemy jungler or to trick them into trying for drag then ult bot to start a fight or secure it for yourself! Just try to be as annoying as possible and balance ganking, farming, and counter-jungling. You can do all 3 decently, so try to be a jack-of-all trades!
PSA You can tunnel really long distances and jump over multiple walls if you tunnel in the right spots, try going into a custom game and tunneling EVERYWHERE to see where the best tunnel locations are. Keeping one end of the tunnel in bushes is really effective. Also, the huge wall in bot and top-lane can be tunneled through for MASSIVE distance straight into the lane. You know the tri-bush-wall just behind the tower? Try heading over there and using that as a great gank-path.