There is nothing wrong with leaving the mosque after the adhaan to go and pray in another mosque

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Check below answers in case you are looking for other related questions:
Ruling on responding to the adhaan recorded on the al-Fajr clock
How the iqaamah is done

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Related Answers:
Is it prescribed to repeat the phrases of the iqaamah twice?
Was the adhaan revealed by wahy or was it suggested by a sahaabi?
Which is more important, repeating the word of the adhaan or hastening to break the fast?
Ruling on responding to the adhaan recorded on the al-Fajr clock
How the iqaamah is done

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