If the insurance company had the damage appraised the appraiser will have obtained a guaranteed price of repairs at the shop of your choice. The check issued should be for this amount. If this was not done, talk to the adjuster. You will probably be asked to sign a release before they send you a check.
If additional damage is found during the repair process the shop will call the appraiser for a re-inspection and “add order”. Any add order charges will be paid directly to the shop. The insurance company will pay the add order as long as it was approved by its appraiser.
If you are dreadfully confused, go ahead and ask State Farm to pay the loss under your collision coverage. You will have to pay the shop your deductible amount when the repairs are complete. You should eventually get it back when State Farm pursues subrogation against the other carrier. Since the other carrier has accepted liability, this should be a “non-chargeable loss” and should not affect your future premiums.
Using your collision coverage is usually the easiest and quickest way to get your car repaired. If comparative negligence is applied to the accident, using your collision coverage is advantageous to you.
I’m not sure what you mean by “most auto repair shops want to go with State Farm”. The shops don’t care which company pays for the damage. State Farm (and other carriers) often select certain shops as “preferred”. That just means that they trust those shops to provide good work and to stand by their work product. They frequently inspect and audit these shops. If the shops do not meet their quality standards they are dropped from the preferred list.
You do not have to go to a preferred shop. You can chose any shop you want. However, a shop that is not “preferred” may not guarantee their work. Chose wisely. Tell the appraiser where you want the car fixed so that he can obtain a guaranteed price of repairs with that shop.