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Hello,

as a European citizen, you can freely travel to any other Schengen States for 90 days.
After that period, you must go to another Schengen States or start the resident process. (legally speaking, reality is a bit different as long as no one know...)

Meaning, you've 3 months to find something to rent/buy.
Some landlords might be scared about your ID card, because they've never seen one from Poland. Showing your passport will fix most of the problem you might encounter, even if it's not needed (legally speaking). Same about administration (bank, insurance, ...), your passport can be proof to be useful.
(Remember that most of the landlord will ask you to put 2 times the monthly payment on a blocked a bank account - rental guarantee - that you'll get back when you leave your rent and if you didn't break anything)


Once you've found an accommodation and signed the contract, you must go with it to your commune (+ work contract / payslip if you already have / health insurance valid in Belgium)
In order to avoid any issue, you should translate (sworn translation) every polish papers into NL / EN.
The commune will verify that everything is in order and the police will ring your bell within 2 weeks, to verify that you're living there (appointment can be done)
==> don't forget to add your name on the mailbox + next to your ring bell.

Then you'll finally receive your "E" id card, European Citizen, valid 5 years. (I guess that's what you mean by Belgian ID, and not the nationality itself/becoming Belgian)

That ID card can/must only be used in Belgium, and is not officially recognized elsewhere in Europe.
Meaning, that you must always have your polish ID/passport with you when you're traveling outside Belgium (==> to proof your identity and nationality)

Thank you so much for all information!