Your instructor is right about your technique being at the heart of the problem. I believe that if you can correct your technique from the "bicycle rider" way you’re doing it now (bending knees and feet moving in a circle) to a proper technique like flutter (straight up and down from the hip), frog (legs move sideways with alternating angles of fin blades), etc. you should be able to move along at a good clip. Another important aspect of proper propultion is getting in and staying in a horizontal trim (flat or almost flat, not vertical or at a 45 deg angle). This will go a long way to increasing you speed by lowering drag and directing your thrust straight out the back, as well as help resist the urge to do that bicycle kick (almost impossible in proper trim).
As for the use of webbed hand thingies, just don’t do it. Here’s why: 1) it take a lot more effort to swim with your hands than with your legs 2) if you are swimming with your hands, you can’t use them for other important things like working your BC, signalling your buddy, finding your backup reg as needed, clearing your mask as needed, fetching things from your pockets, etc. 3) it looks uncool and is a distraction to your buddies (i.e. you will be the diver who cried "wolf" because the others will get used to your arms flapping around all the time to swim and won’t notice if you are trying to signal them that you have an actual problem)
I believe that many, if not most, of us have one leg stronger than the other, with a natural tendancy to drift to one side when not using some sort of visual reference (following features on the bottom, maintaining a compass heading, or just staying inline with your buddy). This can be overcome by using visual cues when available, remaining aware of which way you tend to drift, and angling your fins slightly to the side that compensates for your natural drift. I have that taking this compensation technique to the extreme will even allow me to swim straight using only one leg or if I have lost or otherwise removed one fin.
I suggest working first on your basic trim and propulsion to finish OW class. The rest is something you will be able to refine over time as you put some bottom time under your belt.