Glad you're OK that's the important part!
Don't know whose insurance company you plan on going through but just remember no carrier can dictate what shop you use. Don't give a statement on recorded line to the other carrier, if the other driver denies fault and beats the ticket, even on a technicality you could be in trouble.
If it's repairable choose a shop wisely, like they've been in business a long time with a great reputation (the repair is only going to be as good as the techs that do it), offer a lifetime warranty and the paying insurer has a working relationship with that shop and will back up that warranty.
Adjusters come in all kinds both good and bad. A field appraiser/adjuster is the person that comes out and looks at the damage and if it's repairable negotiates with the shop and handles the towing /storage /physical damage or moves it if it's a T/L. The inside adjuster handles the rental/bodily injury/coverage/liability issues. If that field appraiser doesn't have the check book then the inside person pays the claim. Plan on talking to multiple people and in the end go with whatever company is going to do the best for YOU. Even if you have to go with your own company and pay your deductible they can subrogate against the other to get it back for you. Some companies waive the deductible if both vehicles are insured by them, it's called a cross-claim.
There's a lot to deal with and more if it's a T/L, don't accept any settlement offers that aren't to your liking or in your best interest. Remember, going by what you said it wasn't your fault and now even repaired it's not the same value vehicle as it was before the accident and has a loss history attached to it.
Dealing with an insurance claim can be like buying a new vehicle (sales person/deal closer/manager/F&I person) to maximize the dealership profit. At the insurance company each person you talk to has a specific area of expertise to minimize the payout of the claim (severity).
I used to work for a really large carrier and have settled thousands of claims.