Is it possible to join, go through tech school and apply to the Academy? And would I be able to go even though my contract wasn't completed yet?

Yes, however it is not the common way. The number of prior enlisted cadets is relatively small. Applying straight in, or going through the prep school are better (statistically and time-wise).

Yeah, for PS to go to the academy, there's a FINE line of non waiverable qualifications you must adhere to.