Wouldn’t say that after 23 and a bit years of swimming I’m a master of backstroke but I don’t think my stroke is too bad. You maybe need to tip your head further back if you are having problems sinking and swallowing water (chin too low and your feet will descend towards the bottom of the pool). I physically can’t float in water due to low body fat but I don’t sink on backstroke. Also, keep your head totally still otherwise you won’t be able to hold a straight line in the pool. Quite a lot of pools I’ve swum in have straight lines in the ceiling above the lanes you can keep your eyes fixed on. The unfortunate exception to this is Leeds International which used to have a nice artistic zigzag pattern in the ceiling 
You should try and enter the water with your little finger first each time (brushing past your ear). As you pull your arm downwards from above your head you bend the elbow and sweep inwards somewhere around shoulder level. If you don’t have flags marking the last 5m of the pool then you’ll have to spot something on the pool deck and count your number of strokes from there in to the wall either that or take a quick look by arching your neck as you begin a stroke with your arm. Not sure if that’s a help though there are probably better responses to come from others.
John