What is my legal standing? Can I get them to pay for my home repairs
It is extremely unlikely that they will pay your Katrina claim (this Hurricane happened almost thirteen years ago, in August of 2005), in order to induce you to sign your 2016 insurance forms.
It isn't clear why you would "recently" from the perspective of 2018 make a minor claim from a 2005 Hurricane. Perhaps you just discovered the damage or someone just told you that damage that you knew about was covered by insurance.
They probably denied the claim because it wasn't made promptly or you couldn't prove that it was actually caused by Hurricane Katrina.
They may have seen a claim made this late as a flag for a potential insurance fraud, and hence cancelled your policy because of concern about potential insurance fraud in the future.
(To be clear, I'm not accusing you of insurance fraud. Sometimes minor claims are discovered long after an incident happens, for example, when you inspect your roof many years later for another reason. But, this is probably how the insurance company bureaucrats are interpreting the situation.)
If it was a minor claim from a 2005 incident, your ability to sue the insurance company successfully for failing to pay a claim made "recently" is probably almost nil. The amount isn't economical to sue an insurance company over, and your odds of prevailing are low.
They already have the 2016 premiums in hand so they have no need to sue you to get that money. And, they probably can't be compelled to return the money to you after the policy period is over, just because you refused to sign the form after the fact.
They want you to sign the forms so that they can deny a future claim against your policy for fraud if something in the form was untrue and a 2016 claim was made against that insurance policy. They would care more about this if they feared you poses a risk of making a fraudulent insurance claim against them (again, despite the fact that their fear is probably totally unfounded).
They still care because claims on personal casualty insurance policies typically cover incidents that took place during the covered period (most commercial casualty insurance policies instead cover claims made during the policy period), so a claim could still be made on a 2016 policy in 2018 for a 2016 incident that they don't know about yet.
If you don't sign, they can just refuse to pay any claim on the 2016 policy, at least until you do sign, rendering the 2016 policy useless to you.
They also have no interest in cutting you a break in exchange for your future loyalty to the company because they've already canceled your policy.
So, you should go ahead and sign the 2016 forms, unless you 2016 it contains false statements, so that if someone makes a claim on that policy, it will be paid without the barrier of the fact that you didn't sign the form.