Yes, there are similar modulos to other powers. For example, $n^6\mod504$ can take only the values $0, 1, 64, 217, 225, 280, 288, 441$ over integers $n$.

The $p$-adic methods are used to prove the insolubility for some Diophantine equations. For example:

There are no integer solutions to $$x^5+y^5+z^5+t^5=5$$, as this can be proved modulo $11$.

Also, there only integer solution to $$x^2+3y^2=2z^2$$ is the trivial $(x,y,z)=(0,0,0)$, as this can be proved by applying the $p$-adic method for $p=2$.

However, the insolubility to the negative Pell equation $x^2-34y^2=1$ cannot be deduced from the $p$-adic method.