Hi, whether you join the institutes that you have converted or take another try is a question that has no definitive answer. It is something that you need to decide on after considering a lot of factors. We will try to list out a few of them here-
1. What was your current percentile and what is the maximum percentile that you can achieve(tentative)
2. The programme details of the above institutes - make sure that you specifically check out the placement details, fees, ROI, faculty, and batch profile to name a few.
3. Also evaluate your profile - if it is not decent, start working on it. Check out this video by Sayali Maam for reference:
https://youtu.be/gCT3uhN4w-A
Coming to the second part of your question, a gap year definitely hurts your chances to an extent. But this is something that you cannot change right now. CAT 2022 is still 6-7 months away, and it is advisable to look for roles that are not highly demanding to make sure that you have some work experience. Having a gap affects in two major ways -
1. Most colleges have scores for months of work experience - you won't get them in case you are not working.
2. In interviews, you will definitely be asked why there is a gap, and what effort you put in during this time. Having some side projects/internships/volunteering experience might help you answer this question well.
Hope this helps you out.