Criminal Mischief to the Third and Reckless Driving

My question involves criminal law for the state of: Alaska

I had gotten into a car crash on the 7th of January. At first I had thought the other driver was at fault and felt that I was driving normally.
I had started a claim against against their insurance.

The police began their investigation, and on the 8th came to my residence claiming that I was lying to them and was trying to get money from the other persons insurance. They began to question me, and after that they arrested me for charges of Criminal mischief and reckless driving.
When they had stated they thought I was at fault and brought their reasons for assuming that up, I stated that if that was the case then I would assume liability for the damages and file a claim with my insurance provider to pay for the repairs to the other persons vehicle.

They then proceeded to arrest me without telling me what I was being arrested for, and took me to jail.

They are claiming that the damage to my car was their prior to the accident on the 7th, even though I had the car appraised by a licensed dealer TWO days prior to the incident on the 7th and provided the paper work to prove it. The dealer gave me an appraisal of 3,000 out of a max of 4,100.00. The only reason why they took 1000.00 off was because of the higher miles on the car.

They also think that I Had intentionally hit the other persons car.
Which was not the case what so ever. I was just trying to drive home.

I guess my question is, if I pay for the damages to the other persons car, is there any way they could still throw me back into jail for this incident?

I should also state that I wanted to be represented by an attorney but couldn't afford one. They never gave me an attorney yet. If I haven't spoken with an attorney, can they still press charges on me at my trail, or do they have to wait until my right to an attorney is upheld?

I do not feel that I should have been thrown into jail because I have insurance coverage, and was willing to pay for the damages.

I know I have to talk to an attorney, I just want to know what the possible outcome of this case might be.

There are many things that could happen in two days. The burden of proof is on the prosecution.