State laws do differ but from what you described it would appear that in general what your insurance company is suggesting would be legal. It sounds as if you placed a claim with your own vehicle's collision coverage and thus there would be your normal deductible due, even if you were not at fault.

If you instead placed the claim and it went through the at-fault party's property damage liability insurance policy then you would not pay out a deductible. Since the claim has been made with your insurer, they sound as if they are ready to settle for the actual cash value of your vehicle at which time they would take possession of the vehicle and stop the storage fees from compounding.

If you are hoping to recoup your deductible you should ask your insurance adjuster if your insurance company plans to subrogate with the at-fault party's insurance provider. If they are you may see if they will try to compensate you for your deductible as part of the subrogation procedure.

If you would prefer to go through the at-fault's insurance company for the claim instead of your own then you should discuss this with both insurance companies and see if it is possible at this point in the claims process.