Well, start by giving yourself a pat on the back for recognizing that Alexander Prime should not be facing the party, and another for recognizing that bringing the unstack marker to the Shared Sentence got you killed. You didn't make those mistakes again, and now you know what to look for if your co-tank does something silly like point the boss at the raid (sometimes this is helpful, but you can learn that per fight as well) or is getting low on HP, which is far beyond the awareness of most tanks. Just being aware of your cotank and party is huge.
By and large, players in this game don't really understand why wipes happened until the highest levels of play because the game doesn't often force you to confront wipes anymore, let alone solve why.
If you feel like you let folks down in your party, that's fine, but don't hang onto it in a way that holds you back. You made mistakes and improved. Those players that quit weren't your fault. I didn't see the whole fight, but I'm telling you that odds are they could have done a lot more to ensure victory than they did, and it's the lack of difficulty in the game overall that caused them to be unprepared to be patient and leave, not you. Heck, maybe they just had a doctor's appointment or a kid to go pick up from school, you don't know.
You improved, you got skills that'll serve you going forward, and that's all part of the process. I hope you had fun overall, too, cuz that's the most important part.
As for specific advice, which matter less:
dealing with your pug cotank is always an element of randomness. It sounds like you got the player that didn't communicate or receive at all, this is pretty common. Working to support him was a good idea so that you could try and supercede that player when they did something dangerous like point Prime at the party. All you could have done was be ready to provoke and face him away in the moment if you can't type while playing so much.
The more comfortable you get with your DPS rotation, the easier it will be to watch the party. This is the position you want to be in, and in fact, watching the boss and the party are actually usually more important than your own DPS until you have enrage timers. So don't feel bad if you have to drop some damage to get stuff under control. Practicing on dummies or FATE bosses while thinking about something else is a pretty good sign you're getting comfy.
Sometimes, sound effects help to get your message out of they're doing something dumb or you need to warn people. You can put those in chat using "<se.7>", lowercase, exactly as between the quotation marks. Numbers go through 1 to 16 and are tied to 'system sounds'. I like 7 as I feel it cuts through the din better than most.
As fight design goes forward, even in savage content, generally they offer tanks fewer and fewer ways to wipe the raid and your role matters less and less.
At the very least, the best things you can do are:
hold aggro
avoid unnecessary damage
make sure you (and them) are always top two on enmity ( you can shirk while main tank to keep them in second. This prevents a main tank dying and the boss running loose while the cotank isn't aware and doesn't put on stance/provoke to help)
Reduce damage on whoever's in danger of dying to unavoidable damage(usually players, especially healers and rezzers, that're struggling to avoid damage and have vulnerability stacks), this includes your cotank
and finally, deal damage.