Would I get a new passport or will the correction just be printed on the current passport?
The passport correction or change will be issued on a new passport book.
How long is the validity of the corrected passport?
If you apply for corrections less than one year after it was given to you, you will have 10 years of validity, but if you apply for corrections one year after it was given to you, the validity will be the same as it was on the past passport.
Can I have a passport that was given to me with a limited validity changed to a full validity and with no government fee?
Unfortunately no. You will not be allowed to get a no-fee reapplication service. If you got your passport overseas because of emergency purposes more than one year ago, you would need to pay the appropriate fees for a passport renewal. If you got your passport in the US for limited validity because of a document issue more than a year ago, you would need to follow the steps on the new passport procedure and apply using the Form DS-11. This applies full government fees.