NP-complete problems not "obviously" in NP
There are at least four such $NP$-complete problems listed in the appendix of Garey and Johnson's COMPUTERS AND INTRACTABILITY: A Guide to the Theory of NP-Completeness.

[AN6] NON-DIVISIBILITY OF A PRODUCT POLYNOMIAL
INSTANCE: Sequences $A_i = \langle (a_i[1],b_i[1]), ..., (a_i[k],b_i[k]) \rangle,\ 1 \leqslant i \leqslant m,$ of pairs of integers, with each $b_i[j] \geqslant 0,$ and an integer $N$.
QUESTION: Is $\displaystyle \prod_{i=1}^m \left( \displaystyle\sum_{j=1}^k a_i[j] \cdot z^{b_i[j]} \right)$ not divisible by $z^N - 1$?
Reference: [Plaisted, 1977a], [Plaisted, 1977b].  Transformation from 3SAT.  Proof of membership in NP is non-trivial and appears in the second reference.

The other three I found in the appendix are:

[LO13] MODAL LOGIC S5-SATISFIABILITY
[LO19] SECOND ORDER INSTANTIATION
[MS3] NON-LIVENESS OF FREE CHOICE PETRI NETS