I'm in a similar position but I don't feel the urge. I'll do the Zhongli quest when I'll have some time, which I gather won't happen until 2 weeks or something. I absolutely feel nothing at all while considering this delay.

And, I mean, the quests aren't timed. The bosses drop essential stuff only for the new characters. Nobody is forcing you to play immediately these things.

The only reason you feel this it's because of social pressure. You check this sub and see that everyone it talking about and you feel like you've been left out of the party. That's FOMO.

The truth is that even if you did these quests ASAP, your social interaction would have been the same. You are not being left out.

Take your time. Do these quests in the next month, or when you have the time.

Unless it's a timed event, the quest will be there waiting for you.