If you turn down a plea offer, go to trial and you’re found guilty at sentencing, can the DA ask for more time than the original offer before trial?

Yes, after trial the DA can ask for more time than they initially offered before trial. Sometimes a District Attorney will indicate that an offer will stay open until the next pre-trial however they are not bound by their initial offer. It is quite typical after trial for the DA to request an increased sentence.

Yes, anything up to the statutory max, but it is up to the Judge to impose the sentence.