Move to Italy a bit unsure what to do

Recently moved to Italy with my girlfriend.

We tried living in Sweden for a while, but she didn't like it and we decided to move to Italy, where she has lived for the past 12 years.

We didn't find a permanent apartment right away, so right now we live in a tourist apartment and rent by the month.

She has a job here in Italy (since she got a job we moved here pretty quickly) and is registered in Italy with an address in her old apartment. She already has a bank account, Italian ID card and code fiscale.

However, she canceled the apartment when she moved to Sweden. The landlord now wants to remove her from the address, which is understandable as she no longer lives there.

I work for a global company and can settle anywhere in the world. Right now I am in the process of moving my contract to the Italian entity of our company. Then I will also have an Italian contract.

I have already arranged a code fiscale.

Now to my questions:

- I have problems opening a bank account as they require an address, I cannot give the address of a holiday home as it is not a valid address according to them.

Is this really right? Even when I have Italian income, surely I must be able to open an account even if one does not have a permanent residence?

- soon I've been here 3 months and as it looks now, we won't find a permanent home any time soon, but will have to continue renting Airbnb or tourist apartment.

Can it affect my right to stay in Italy and live here as an "Italian" with the right to health care, pay taxes, etc. here?

A bit unsure of what to do in the situation as we both risk being without Residence after the end of the month, however we both have Italian contracts, work here and we have somewhere to live even if it is not a long term contract or owned by us.

But want to start life for real, to be able to pay taxes, have a bank account, the right to healthcare, etc.

Right now we almost feel like tourists.

To add is that we are both EU citizens.