From the photos, it appears that You have two separate but inter-related problems. 1) teeth that are not aligned (not "straight"). 2) A bad bite where your upper teeth are to far in front of your lowers and most of your teeth don't touch (an open bite). this is a very difficult combination to fix and it is not as simple as extraction vs non extraction. If it was done none extraction, your bite would get much worse and need jaw surgery to fix. Extractions with braces would make it possible to both straighten the teeth AND make the bite better (but probably not perfect) but avoid surgery. There is a 3rd option using TAD's but that would still require extractions. If you want to get this fixed, find someone (an orthodontist) really good who does not promise a simple solution and runs through some similar choices that I have just touched on.
