Edit: Sorry, this turned out a bit long. But I feel for your dilemma, itās near to my heart. c:
This may sound crazy, but have you tried joining guilds that are newly established? They tend to have fewer people who already know each other.
Thereās also something to be said about sticking with it even if you donāt fit like a glove right away. Any guild worth a damn will expect you to fail at first. Ours for example takes in what people would call ānoobsā; those whoāve never raided before. We know they havenāt, so of course we donāt talk bad about them when they mess up mechanics and such.
In the same vein, any guild worth their salt will also know that new players need to be integrated into the group just like in any social setting - it doesnāt happen on its own or overnight. If a guild expects you to just fit in automatically without any help, theyāre in the wrong not you. They have the numbers advantage, they should be helping you settle in.
I get it. Iām actually very anxious too. Iāve played WoW for so long that itās like a second home to me, and thatās the only reason I feel less anxious around its players. Thereās a certain ebb and flow to guilds - even the ones that seem like the āglueā of the guild eventually take a long break and yet it might not even have an effect, because someone else will take their place.
Give the guilds a longer try, and make a small effort to get to know the people in it. If itās hard and no one talks to you or tries to take you with them, donāt waste your time and move on. If they reciprocate, give it time! Youāre bound to make mistakes, everyone does. But in a good guild that treats each other with respect, theyāll just tell you and then everyone moves on, wiser and more closely knit.
Finally, about that mistake that wiped the raid⦠those happen. Everyone should have the right to make those mistakes while learning the fights. Weāre stuck on heroic Denathrius. Weāve wiped so, so so many times due to dps failures, because the first phase is only really hard on dps and healers. Tank difficulty comes in phases two and three. Lo and behold, the first time we hit phase 2, tanks immediately made a mistake and wiped the raid. Weād been stuck on P1 for so long, youād think everyone would be fuming at the tank who was understandably mortified⦠but no. Of course not! We all understand that tanks and healers have every right to make mistakes. Nobody thought much of it, in fact it was joked about because it just shows that nobody is perfect, not even tanks! 
Chin up, friend. Anxiety sucks really bad, but this is actually a great game at combating it and helping us overcome some of our social anxiety. I used to be so anxious I couldnāt queue for a random heroic! Now I frequently lead normal Nathria pugs like itās no big deal :'D