I have an appointment for road rest (class C) with Carrollton MC. As I have cleared road test do I need to make a new appointment to submit the paper?

Yes. You need appointment.

For most of the people, First you need to take appointment for an online test. Then, assuming you pass, you have to take another appointment for road test but make sure you take appointment at atleast a month in advance because there is a huge backlog. Take your video certificate with you and practice around that Center itself. Parallel park practice on weekends in that center. But for you, I am assuming you have gone thru the driving school route and have all the docs, you don't need road test. But yes appointment is still needed.

I’m not sure, but I believe you have to because these days everything is working according to the appointments. It’s better if you ask on the front desk.