You actually need only one social guild.
Once you find it, you could mute the chat from all other guilds and focus only on chatting with that one guild.
The other 4 slots you can fill with trading guilds that you can use for selling items.

Take a look at each guild you are in.
Do they frequently change the Message of the Day?
What do they have written in the About us?
Do they have a trader hired?
Is the chat active?

If you can single out one of those guilds as better than others, you could decide to focus on that one guild.

Another approach would be to look for a guild with similar minded people.
Do you like to roleplay?
Do you like to collect achievements?
Do you PvP frequently?
Is English your native language?

If your native language is, for example, german, then you could search for a german guild where everyone will chat in your language.

Don't hesitate to leave a guild that has nothing to offer to you.
If you entered that guild via a random invitation, it is most likely not worth your time.
It is YOU that should be the one asking for an invite, not the other way around.

Another way to approach socializing is to do group content with random people.
Apply for a random normal or veteran dungeon.
Ask for advice.
Depending on the answers, send friend invites to people that seem friendly and are willing to help.
You can then ask those people if they have a "native" social guild, and if they could invite you.

Playing an MMO with friends will is definitely a whole new experience.
Hope that helps!

The criticism is fair and I hadn't made enough effort on the social side.

I looked at my guild membership for the most active looking guilds. I left the guild chat on but it didn't seem like anyone was online. I am probably better off proactively finding a guild. I may also do better on the EU server. That said a bit of transatlantic friend making would be nice too.

It probably hasn't helped to be using the NA being quieter at the times I am on to mean less competition when harvesting nodes.

I suppose some social anxiety can apply. If faced with a conference with crowds of people but not knowing anyone it can be tricky approaching strangers at random to make friends.

A crowd of Avatars, non of whom you know can be no less straightforward to start. The flow would well parallel real life. Listen to discussions and try to join in with ones that sound interesting and hope to built a rapport that way.

This is my first MMO (unless you count Second Life) so there is a learning curve. Prior multiplayer has tended to be more LAN party style where I already knew the other people.

With Second Life the main socialising came after finding a common interest group and I expect the same could be true in ESO hence my best prospect is probably being in the right guilds.

It has been tricky that most of the time when online I am engaged in some specific task so that running a dungeon may mean interrupting what I am doing.

The general quests are not really designed to be played together. I can't invite some to properly join me on a quest (unless it is a multiplayer dungeon). They can be around and kill any hostile enemies that appear while doing the quest.

If I am gathering resources from nodes then it can be a disadvantage being around other players as we will tend to compete for resources and will all be slowed down.

I have spent a bunch of time in state a) and had enjoyed playing anyway but it feels like it defeats the point of an MMO to play it like a single player game. Secondly, particularly while skilling up crafting I realise that working with others has distinct advantages.

The Guild thing I really haven't mastered. When in a group I can look up where each of the group are and their online state etc. With the Guilds I get the notification posts but don't have the same visual feedback about who is online and where they are / what they are doing to use as a cue to join them. I accept I am still learning how to use Guilds properly and it wasn't that long ago in the scheme of things I figured out how to use the Guild stores and started making use of the facility.

The way experience is divided in group dungeons is still a bit of a mystery. I would have thought given it is usually a lot more difficult than grinding mobs that it would be a good way of getting experience but I have run dungeons with (usually higher level) other players and at the end not noticed my experience bar moved much. On the plus side I have usually walked away with some really good gear.

It sounds like you would be a good fit for a social guild, OP. They want to recruit people who want to be there. Trust me. I used to run one. We were always actively recruiting, looking for people who wanted to be part of our small online family. Once we'd get new members, we'd actively try to get them involved and make them feel included. This usually was along game lines: running dungeons with them, helping them craft gear, etc.

I'd like to share a little bit from the other side though. I think you might realize that it's not really you. ESO is not conducive to social guilds. There really isn't enough meaningful content in this game that makes people actually have to work with others. Grinding to max level can be done by yourself. Dungeons, though optional, can be done without knowing people by using group finder or just joining a Pug group in the undaunted area. Really, the only content that is pushing towards social guilds is end-game content like trials, which are a relatively new thing. Unfortunately, it's so end-game that the people who are likely to be interested in a social guild (the more casual players) aren't even thinking about completing this content.

Because of this, finding a good social guild will be difficult. They can only exist if the leadership goes against the grain to keep them alive. They do exist though! I think you'll find that once you're one, they'll help you feel at home.