You are most likely missing one or two email parser setup related steps. Please read the 2nd post from here and it should help (most people forget to match the from email address field value with the AWeber parser):
Also, go through these forum posts which might help:
AWeber – https://support.tipsandtricks-hq.com/forums/topic/aweber-integration
AWeber – https://support.tipsandtricks-hq.com/forums/topic/cant-get-aweber-working