The prosecutor can offer different pleas as many times as they want. There is no certain limit. I would certainly not take the first offer and would challenge the evidence and plea. Sounds like you need to speak to an attorney so that the prosecutor knows you are serious about getting the best plea and challenging the evidence.
A plea bargain is just like any other negotiation. Every prosecutor is different. When you buy a car, some dealers may give you their best price upfront, others may go down in steps. They don't want to lose the sale, but there is also some point below which they won't go. Similarly, the prosecutor doesn't want to force the case to trial, but there is some point below which he won't go. The only way to tell is to know something about what is standard for this offense, etc. That is why you should have a lawyer helping you.
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Your questions demonstrate the need for you to have an attorney. They are matters for consultation with someone who has your interest but is also distant enough to see the trees and the forest.
The above is not intended as legal advice. The response does not constitute the creation of an attorney client relationship as this forum does not provide for a confidential communication.
You should hire an experienced criminal defense attorney from the county where you are charged. They can look at the evidence and tell you whether, in their opinion, the State can prove it. The State has no obligation to make an offer, and can withdraw it at anytime. When I was a prosecutor, I would very often make one offer and withdraw it after to many continuances. You may be able to get another continuance, but you should really get yourself an attorney.
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