Car damages caused after handing over to the Insurance Company for claim process and repair

On 8th of Feb 2016, my car was bumped from behind. I called the Emergency number for towing assistance and reporting accident on the phone number provided by the insurance company. A1 Assist was at the spot and spoke to the insurance emergency line. They drove the car instead of towing until they are offsite of me, not sure if they towed it later from some place to yard. The claim was processed after nearly 3 weeks, and the assessment said the damages are to the front and rear bumper and a crack to the radiator. After that it was sent out to the company's service center (not sure if third party) on around 4th of March 2016. The person from there called me and said the vehicle should be ready by 3 weeks or max 4 weeks. After 4th week, I got a call again around 8th of April 2016, saying that they are trying to start the car but seems there are problems in the engine. I asked him to inspect. At around 14th April 2016, the service person called me again saying there were damages to the Gasket, some coil and Valves and will incur additional cost which I must pay in addition to the Excess Amount. He was very sure that happened due to car being driven after the accident and damage caused to the radiator. I followed up and I was said by the assessor that it was due to wear and tear of the car and tried to blow that on my car's maintenance. I explained him the fact that the car was major serviced on 6th of Feb 2016, two days before the accident, and all the wear and tears were replaced and had no such issues with the engine. At around 20th of April 2016, my complaint was redirected to his manager for escalation and the manager said he will investigate. And upon investigation the A1 assist accepted that they drove it for 100m from the accident spot and then towed to yard, but need investigate who drove it from there. I am without my car for almost 3 months now (the premium is debited though each month) but there is no conclusion on who must pay those damages. Clearly after the car was handed over to the insurance company, anything that happened to the car, they must take all responsibility and do their investigation in the background. I must not be held responsible for those damages. I was told to sign the release at the service center after inspection of whatever initially the assessor mentioned damages were worked upon. But the assessor did not consider the damages caused by them (Insurance company), and if I sign the release, the service center will ask me to pay the additional cost whenever the car will be ready (which he said will take another month). I am awaiting the Telesure to come back on this, I have been spending so much on rental car and Taxis and phone calls to follow up on this complaint for nearly three month now. Highly frustrated!!!