The clear answer is: No you should not have this professor as your advisor. Why? Let's take a look at what you know so far.
Last semester, I took class A and loved the material. It was a subject I excelled in and professor A' was someone I understood well.
You liking and excelling in a subject or a course or the material is totally orthogonal to whether the one teaching it is a good teacher. There were quite a number of courses I took where I got the highest grade but that I felt were terribly taught, because I could see that the teacher was simply unable to teach well. Everyone who was already good enough didn't need the teacher, and everyone else who needed help couldn't follow. In some cases the teacher was trying his/her best, so I'm confident they would eventually become good teachers, but in other cases the teacher had a wrong attitude toward students, sometimes one of utter laziness and sometimes even one of condescension. I don't need to tell you to stay away from the latter sort.
I think it is likely your impression of professor A' was mistaken in some way or another.
I thought it was perfect to take the course 2A and make A' my thesis advisor, but this entire semester, he missed many classes (due to sickness and/or being the Chair of our department), was only available for 'pop in' office visits, rather than scheduled ones or scheduled office hours, and we rarely had homework we could do, because our class was riddled with confusing theory and no examples.
Missing classes due to sickness is not at all his fault, but neglecting his teaching responsibilities due to being department chair is simply being irresponsible, unless you think the department itself has unreasonable demands. However, I would have expected that whenever a professor is sick another professor will temporarily take over the class and continue teaching the material as best as possible.
This judgement is further supported by the fact that he does not put aside office hours for his students, does not give homework for practice (not to say go through the solutions), and gives no examples. It would however be necessary for you to find out whether your classmates in general felt the same way, to ensure that your confusion over the material is not because you lack some prerequisite knowledge.
For a grade, he asked us to learn and present material in class. I'm a student who needs to bounce ideas off others in order to really understand material really well. Since no one (fellow grad student wise) could help and he was always unavailable, I stuck to books and mathstackexchange. I got turned around many times
It is sad that no fellow student was willing to help you. Did you ask him when you could see him (instead of asking whether he is free at certain times)? If so, then he ought to try to help you!
and when I had to present, he constantly interrupted me to tell me "what [I] meant to say". After 40 minutes, I got so worked up I started to tear up.
Whether this was bad teaching depends on exactly how he said it. A teacher who is very meticulous might constantly point out slight inaccuracies, but in a way that does not put down the student. But most teachers do not know how to do this. Even worse, some teachers put down students simply to show off. But the question is about you. We don't need to know or speculate about your professor's motives. If you cannot take his style of teaching, don't force yourself!
I felt like the time I spent trying to work through material by myself was worthless, and anything else I could say wouldn't be "up to snuff" in his eyes, but I pressed on through hiccups and tears. I know he was trying to make me a better student, but I'm not sure I can take a year of trying to do something right on my own, rarely meeting up with my thesis advisor, and falling flat during my thesis defense.
Indeed, even if he doesn't mean it, his attitude or style is very likely to be unsuitable for you, in which case it makes no sense to not look for a advisor that is better for you!
Is this just a personal problem, and I should consider counselling?
Of course you should also consider it. As long as your university supports confidential counselling, it may be a good thing for you to go for it, not because there is any problem with you, but because they might be able to provide all kinds of help (not just a listening ear) that you surely cannot get from Academia SE or elsewhere on the internet!