Assalamu alaikum ustadh. If I wake up like 13-14 mins before Fajr ends, if I try hard I could maybe finish wudu by 7-8 mins but I didn't I did it in normal pace and ended up missing the time, am I sinful?

You are sinful if you were heedless by staying up late for no valid reason, not doing the necessary means to wake up on time, and not hastening to pray on time instead of taking your time until the time is over. Therefore be careful and do your best to observe the salah at its proper time.