No. Your answer has absolutely nothing to do with infinity nor is it paradoxical.

The paradox is saying that for every number N there is an occupied room in the infinite hotel. But that there is still vacancy.

That's it.

We take every guest in their room N, and ask them to move to room N+1, and now room 1 is vacant.

So the infinite hotel is always full but always has vacancy. That's what makes it a paradox.

The important feature of Hilbert's example is not the hotel, or the guests, but the strange consequence of Cantor's method for 'counting' an infinite collection of objects. William Dunham's Journey through genius does an excellent job with Cantor in his final two chapters. I'm paraphrasing him.

Suppose you want to know if you have more than 5 raspberries. You could count them, or you could stick them on the ends of your fingers. If you have berries left over, there were more than five. If all the berries fit all the fingers, you have put your fingers in a 1-to-1 correspondence with the berries, and concluded that the numbers of each are the same. That's how small children count, and Cantor's genius was to extend this idea to sets of infinite objects.

If you can find a 1-to-1 correspondence between the counting numbers (1,2,3,...) and some other set of objects, then the size of the two sets must be equal (in the business, size of sets is called the 'cardinality'). It is traditional to denote the counting numbers by 'n'.. Hilberts 'paradox' arises from Cantors claim that there are as many counting numbers as even numbers. The correspondence is n->2n. For every counting number, I can double it to find every even number. That's weird.

You can just as easily show that there are as many odd numbers as counting numbers. The scheme is more complicated, but you can also find a 1-to-1 correspondence between the counting numbers and all the fractions between 0 and 1. It seems like the counting numbers can count everything! But that's not the whole story. The big shock is that there are more numbers between 0 and 1 than counting numbers. That is Cantors 'nondenumerability of the continuum'. I highly recommend Dunhams book. The proof of the nondeumerabilty theorem is so beautiful and accessible (like accessible to a ten year old).

No, because if you have a room empty, you now have to identify the vacated room, which could take an infinite amount of time

Also, it would take up to an infinite amount of distance to walk to or from the front desk, so we just be dealing with immortal guests who have no time sensitivity at all

Furthermore, the room must be made ready and guests tend to check out at a certain time of day, so you would have to decentralize everything while accounting for light speed limiting the speed of information

Some of these problems are solvable if you add more constraints to the puzzle. Like, if you have a rate of check-in to checkout you could do something with statistics and discrete math to divide the infinity into an infinite number of sub hotels, and in doing so possibly guarantee availability for the guest in a long but linear amount of time

It would be an interesting programming/math puzzle. My intuition tells me it is solvable, but it might not be

I've seen the paradox mentioned a lot but I don't see it as paradox at all.

A hotel with infinite rooms has infinite size.

Just idea of asking guests to move would take infinite time (i.e will never happen).

if each guest wait for the his new room to vacate, no one would move at all.