Where can I find records of old FAA registrations?
NC, NR, NL or NX are not necessarily older aircraft. Any aircraft more than 30 years old can use NC instead of just N.

So, to find the a/c registration you just take out the C.

I own a glider that was built in 1965 so my numbers are NC and 5 digits.

It should be the same for R, L, and X, which are any aircraft that has a restricted, limited or experimental airworthiness certificate can use those letters after the N. See FAR Part 45.22.