The Desire to Go Back To My Home Town

As is usual with other schools, there are still some restrictions inat my university due to COVID-19, even though domestic travel has never be recognized en made illegal.

I didn't like "as is usual with other schools", but "As is usual by now, there are still restrictions" would've been fine. Not really sure why, but it sounded bad to me.

And they have been in effect since last summer.

There are some old friends who I really want to see and talk with in person or, and many places wherethat I'm eager to visit.

You're eager to _visit a place_ -> it's a place _that_ you're eager to visit.
You're eager to go _to_ a place -> it's a place _where_ you're eager to go. (Or it's a place that you're very eager _to go to_.)

I don't know the name of the rule about this, but that's how where/that works in a sentence like this.

So I hope I can go back there without hesitation as soon as possible.

"without hesitation" didn't seem to fit, and I didn't know what you wanted to express with it. You could've said "When I get the chance, I'll go back there without hesitation."