Is it OK to change any polynomial time to another polynomial time without breaking equivalency of Turing machines?
Yes.  You can replace the oracle with subroutine calls to the polynomial-time implementation of the oracle.  Or, to put it another way: $P$ is closed under composition.
The running time remains polynomial: if the original Turing machine used $p(n)$ steps, and the subroutine takes $q(n)$ steps, then the resulting machine will use at most $p(n) + q(p(n))$ steps, which is upper-bounded by a polynomial.