As you mention, there are a couple of ways to handle this.
What's most important is that you chat with your group and they agree.
The rest of this answer is largely opinion and personal experience.
If your character doesn't level up, that puts you in a worse position than the other players when you rejoin the game. This doesn't exactly make people enthusiastic about running the game, in my experience.
"So you put in the hard work GMing for three sessions, enjoy being a level behind forever!"
The idea that because your character was not at risk they should not advance has some merit, but I think that because you waived the opportunity to gain the advancement so the others could play you deserve it.
Obviously, this has a little to do with play-style, if you're playing a hardcore "if you die you restart at level one" type game, then this carries less weight.
As a GM you have a lot on your plate. You should, you need to concentrate on running the best session for your players, balancing focus, keeping play flowing, dropping clues and plot hooks, tracking NPC knowledge and agendas.
Adding in your PC on top of that, as CatLord pointed out, is just asking for horrible information leak, as well as a split focus that can be very harmful for the game.
Even having your PC fade into the background can be dangerous - I once had my PC accidentally derail a whole bunch of plot threads because another PC asked them to do something they had no good reason to refuse and it spiralled out of control, dragging them further into the plot.
If this means your party hasn't got a rogue, for instance, cut down on the trapped doors. Simple. If the adventure was likely to need one they'd have brought your character, after all.
Influencing the Story
As GM you already influence the story. Write a story starring the other PCs, give them the limelight, and let them do the same in turn. If you aren't running the game for the other PCs the players are going to have trouble engaging with your game
In my view your character's plot arcs are better handled by someone else (though you could possibly drop some hook, but if you do be sparing with them), that way you can enjoy the exploration and discovery modes of play.
My advice would have a mission-based format, possibly from a higher authority.
This way you can have a rotating cast (the ones who are right for the job), excuses for missing characters (doing another task), and keep interplay between adventures run by different GMs minimal if needed.
Also, drop loads of plot hooks, even ones you don't plan on using. If other GMs want to claim them, let them, if not drop them or link them into your main thread. At worst, someone else can GM for a bit while you figure it out.
This kind of game can be a lot of fun, but there is a tendency to tread on other people's toes. Talk to the other GMs about what is permissible, what they should ask before doing, and when surprises are OK. Also, be flexible and willing to rewrite something when another GM does something that would be contradicted by it.