Carolina rig: personally I rig mine with a 3/4-1oz cylinder style weight, glass beads (2), swivel, and then about 2' of leader to a 3/0 hook. My FAVORITE all time lure on a c-rig is a black zoom trick worm. But pretty much any plastic will do GREAT on a c-rig. Trick worms and the like are great in clear-stained water, and I like creature baits and beavers in dirtier water. I find the c-rig is best in 20FOW or less, and a heavy football jig or drop shot seems more productive deeper than that. To fish, cast out to submerged humps, offshore bars etc, and drag it back with a sideways sweeping motion. Don't pivot too much at the waist, only pull the bait about 2'-3' at a time with your rod. This will keep you from being out of position when a fish strikes, as your hookset should typically be reel down to the fish and then a solid sweepset. Feel the bottom and make a mental map of what is happening down there. When you hit a piece of trash, shake a few times to click your glass. Then ease it over and continue on. You can occasionally lift your rod UP and work it like a t-rig, but doing it that way constantly hops your weight back to you and you lose contact with the bottom too much. Anytime you're throwing a c-rig, its good to have a jig and a crankbait that will dive to the depth your fishing nearby. This will change things up if your school stops biting, or if you find something on the bottom you want to probe more carefully with the jig.
Your bite seems like its always either at the end of a drag ( that's why I try not to be out of position) or when you are shaking the glass on submerged cover. Alot of bites will be pick N swims, so watch your line. Other times it will be just like a t-rig with a solid thump.
Alot of people like to go with braid as a mainline on their c-rig with say 40# braid to a 12-14# flouro leader. This is super sensitive and will translate the bottom to you very well. I did that for a while but found that my lead got hung up WAY more than my hook did, and so I switched back to. #15 yozuri ultrasoft and a 12# flouro leader. This lets me break off easier if I need to... But I feel still gives me more than adequate bottom sensitivity.
Ymmv.
As far as a dropshot goes, there are far more qualified people on here than me to advise you on that technique. I have used it time to time with fair success, but in the waters I fish, I find myself always going back to a c-rig. I like a trick worm, roboworm, zman finess worm or a senko as my drop shot baits. Same as what I use as a shaky head.
Good luck! Hope I helped a little.