I need more information. Has the State filed charges yet? You stated they are still investigating the case? I am a former prosecutor from Broward County and can assist. You can call my office at (954) 926-0202 to get further assistance from me.
When you were arrested, the mug shot company pays to get your mugshot and put that on their website. It is a public record. In order to get this removed from all websites, there are services out there that may assist you in having it completely removed from the internet. It can be costly, usually around $500.
you need to seek the advice of an experienced attorney. I would ot accept the first offer from the prosecutor. An attorney should be able to get you a withhold of adjudication and prevent you from doing any jail time. You can always ask the judge for a withhold of adjudicatio, but most times when a judge sentences you to jail time, then there is a conviction, unless the jail is a condition of probation.