All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.
If you are certain that you are performing the prayer after its prescribed fixed time has started, then your prayer is correct. Nonetheless, you should know that it is an obligation to perform the congregational prayer in the mosque as clarified in Fatwa 82538. So, a person who prays in his home or his office and does not perform the prayer in the mosque with the congregation without a valid reason, he misses a great reward and is sinful for not fulfilling an obligation of performing the congregational prayer in the mosque.
Therefore, you have to be keen on performing all the prayers in congregation in the mosque as by doing so you get a greater reward and you please Allaah, especially that you mentioned that there are many mosques in the area (where you live). The difference of the time of the Athaan from one mosque to another is not a valid reason for you not to perform the prayer in the mosque. You have to choose one of the mosques which you are certain calls the Athaan according to the prescribed time and perform the prayer in it.
For more benefit on the times of the prayers, please refer to Fatwa 82907.
We also advise you to earnestly continue reciting the Quran. Reciting the Quran is one of the best acts of worship by which one seeks to get nearer to Allaah, especially if one ponders about the meaning and recites with attentiveness.
Allaah Knows best.