Do we have to accept the insurance company estimate on damages or are they obligated to pay us based on our written estimate on furniture damages.

Neither.  If you can't agree with the insurance company on any aspect of your claim - how much your furniture was worth, how much their value was reduced or how much it would cost to restore or replace the furniture, how much of your damages are covered by the policy and how much be a deductible or other reduction, the extent to which your damages were caused by a covered event versus cause that is not covered, etc. - you will have to assert your claim in court (or arbitration, depending on what the policy provides) and the judge or jury (or arbitrator) will decide the issues.