Rope has no bearing on you coming out of the water at all. It does make it somewhat easier if it's on your tower vs being properly placed on your ski pylon. Your issue from what I am reading is probably 95% technique and 5% driver. Yes the driver does make a difference but if you can ski it shouldn't make a huge difference.
You don't say what foot you ski forward with? A simple reminder that I like to give kids or newer slalom skiers is ski placement right before resistence force is being applied (boat pulling you forward)
Left Foot Forward - get ski pointing to 10 - 11 o'clock position
Right Foot Forward - get ski pointed to 1 - 2 o'clock position
It's natural for the boat to want to pull your ski into you which causes you to get off balance. Once the ski is pulled under you during pull out you cannot get it back away from you and thus you're defeated.
Also as stated above keep your knees bent too.