Sylfana

You need to check with the department of licensing for the state you’ll be residing in about the validity of your Polish driver’s license.   Here in Washington, my husband’s foreign license was only valid for 30 days, and then he had to take the tryst to get a Washington license.

Moved from K1 Forum to Moving to the US and Your New Life in America - IDs/Driver licenses are discussed here 
 
Every state has its own rules regarding this. 

My wife needed her home country ID to conduct some Home Country financial matters after moving to the US.  Other than that, she hasn't needed it here.  You might need it when traveling back to Poland.  
Driver's license requirements vary widely by state.  In some states, you might have to take a written test as well as a driving test to qualify.  In other states, you might be as fortunate as my wife.  She was able to surrender her Taiwanese driver's license for a fully-valid Texas driver's license without taking any test.  Texas has an agreement wit Taiwan.  Good luck.

If you are talking some sort of National ID, it won't be required in the US. Regarding DL, i'd say renew it and bring it over. Depending on the state, you may be able to use it for a month or two. When I came to US, my state didn't recognize my DL but the insurance company did and so was able to get the heavy discoun for the stated driving experience.

Also look into applying for an International Driver's License while you are still in your home country. The acceptable validity period for IDLs may be longer than 1-2 months, depending on the state's rules.