I strained something in my right lower back while I was trying to lift a heavy cart. The shooting pain stopped me in my tracks. I have been taking ibuprofen and Tylenol and the pain becomes dull but still very uncomfortable and have been increasing the doses. I have tried heat and ice pack also. I made an appointment with my doctor but it 4 days away. What should do? Is this normal? Its been several days

Yes, it's all too common, and it can take weeks to recover from.
Ice is only for swelling. You need HEAT. Even if someone can feel a muscle that is taut, that is different.
Soak in tubs and showers. Heat heat heat. Stay warm from head to toe.
Be sure to move every 30 minutes or so, if sitting. (On a heating pad, of course!)
Move around gently, bending side to side and front to back, just a LITTLE, to try to loosen the taut muscle.
If you have someone living with you, have them feel for that muscle and extremely gently massage it, at each end as well as the middle.
Take more ibuprofin than Tylenol. Neither is good for your kidneys, but Tylenol is worse.
Sleep with as many pillows as you have, behind your back and butt, between your knees, under your arm, to keep you about 2/3 way over on your back. While on your side, do sideways pelvic scrunches.