Contractor concern regarding insurance claim

My house had a water intrusion from the top edge corner of a poorly flashed dormer which ultimately caused my fascia to rot causing a section my ceiling in my living room to collapse. Insurance denied my roof citing "neglect", and determined roughly $3k of damage to the interior of home but below my 5k deductible. I signed a work agreement with a midigation contractor supposedly specializing in water damage.
Hired hands then came out and ripped down a larger area of my affected ceiling to allow it to get dried out. De humidifiers were placed and the owners of company came out same day to survey area to help them write up an estimate for the "mitigation" and repairs.
Two days passed and on a Friday they said they would have an estimate written for insurance to review for approval above my 5k deductible by the following Monday. Monday came and went with no word. Tuesday, they gave more excuses for delay. Thursday, received a text simply saying "done", then they mentioned I hadn't paid my deductible, even though I hadn't received a copy of their proposed estimate. 2 hours later they left a voicemail while I was at work mentioning again I hadn't paid my deductible yet, and need to do so, so they can submit the estimate. They drag their feet with me, until this time, then it's panic mode for funds.
Feeling uneasy about their approach and gaslighting tactics, I called my adjuster to see if an estimate was submitted, he said he didn't know of one, said you should receive it first. So talked to contractor and asked to have them tell me what they would charge for me to walk. They said it would be in my financial best interest to continue. They said it would be 4k to walk. Asked them AGAIN if they submitted an estimate, said yes, sent to claims department, but when I asked them to send me a copy of estimate, as well as itemized invoice from services rendered if we parted can ways, they didn't produce. They said they could take me to court for a full 8k, and I would have to pay for their lawyer fees, (I believe they are referring completion total). Was told a contractor cannot sue for work that hasn't been performed yet, but only for work completed, is this true?
Considering offering them 2k, with a signed lein waiver or we can go to court. Thoughts?