Haven’t met any other interns a month into work. How to introduce?

I have been doing this CS hybrid internship at a large company’s HQ for a month in and haven’t met with other interns mainly because I don’t want to go out with any interns after work. I wanted to maintain a rather neutral demeanor rather than boring but obviously that didn’t work.

First week was fully remote and I didn’t make any effort to go to the events in-person and lunch meetups even though I live close by due to laziness and social anxiety. Starting the conversation is what bothers me but I can hold the convo pretty well after starting.

I am the only intern in my team so I have good rapport with my older co-workers and manager.
Each team has one intern and there are 3 other teams near me with 2 guys and one girl.
They all know each other but aren’t super close.

There’s little chance my co-workers/mentors will give any negative feedback as I chat and deliver on time. My manager probably seen that I’m a bit non-social in-person but he’s not extroverted either.

I came across the girl intern a few times walking past me looking at me and I also noticed her staring at me while I was being all talkative with my older co-workers after a meeting and also made eye contacts on other occasions.
I feel bad for her and the others and want to fix it next week before it’s too late.

I am thinking next week go up to her desk and the other 2 guys and introduce myself by:

“Hi [Name]! I wanted to introduce myself without being awkward but that wasn’t possible so here I am. I am [My name] and …”

So what’s your advice?