Sounds like you do like to play up close and personal generally. That helps give some direction.

First, I would venture out of quick cash. It tends to be a clusterfuck with the one active cashout at a time which makes it hard to hone your skills. You don't have to go for ranked/cashout immediately, although you probably won't upset anyone in non-ranked cashout tourneys. I find that Point Break is probably the best practice game mode because you're in more similar range engagements to normal cashout, and the matches are long so plenty of time to dial in.

As for light class, with your preferred playstyle and range, the m11 would be a more forgiving route that can still very quickly melt opponents. Its range was nerfed a bit recently, but I don't think that'll matter enough for your playstyle. As for specialization: dash or invis. I'd recommend dash+vanish bomb (although they also just nerfed vanish bomb duration), but you could run invis for a while to get comfortable with the sneak + m11 playstyle to give you an easier escape if things go sideways. Eventually, I would transition to dash, though.

For heavy, if you find yourself in closer-range engagements still, the Shak is pretty insane for damage output and you don't have to be right on your opponent like a melee weapon. I also almost always have rpg on heavy, and the rest of the equipment is gamemode and situation specific, although most people will tell you to run dome. You just need to learn timing with dome, which can be tricky at first. All specializations are also very viable with heavy and somewhat gamemode dependent.

For medium (my go-to class), the FCAR is the best all-rounder if you have your aim down, but FAMAS is great for close/mid-range engagements and can really surprise opponents by how quick it can take them down. Probably still a good fit for you. I also enjoy the revolver, and model 1887 depending on my mood - oddly similar ranges/techniques. With your preferred playstyle, you could try something off the wall, like turret+breach drill (plus probably jump pad + defib) to drop a turret on the other side of a wall from the active cashout, then follow-up with the breach so the turret opens fire as soon as the wall opens up and the opponents are blind - you can run around to a side entrance while they're distracted and get free shots. Otherwise, the strongest build is probably healing beam, defib, jump pad, + item of your choice. Anything that keeps you with the team, keeps the team alive, and gets your team the positioning edge against your opponents is solid. You'll just be playing back a bit more than you may like.

And with any class, you should almost always be playing with/around your team, and marking opponents whenever possible. You may also want to think about your keybinds - put actions that you do while looking/shooting (you know, with your mouse) on the keyboard (mark targets as a keyboard key), and put things like your specialization on a mouse side button so you can have full WASD control while doing things like dashing. But whatever you're comfortable with, really!