Is it permissible for me to pray Fajr immediately after the time of imsak and before the adhan, due to the difficulty I face from the adhan being delayed by approximately an hour after iftar in non-Ramadan days?

If the time of Imsak (abstinence) is at the advent of true dawn (Fajr Sadiq), then you may pray after this time has begun, even if the adhan (call to prayer) has not been recited. However, if the adhan is what determines the advent of true dawn, and the Imsak time is merely a precaution, then it is permissible to eat and drink after the Imsak time, as long as true dawn has not appeared. In both cases, it is sufficient for you to wake up before dawn with enough time for suhoor (pre-dawn meal) and prayer.