I commend you as the homeowner for recognizing the problems/causes,it will go a long way in avoiding encore or repeat performance. I can't stake my bank account without on site inspection but I believe you will do best by removing the entire cover to allow permanent repair of damage to main structure then construct cover FREE STANDING . I have asked contractors and homeowners why they insist on supporting one side of a patio cover from the house and they just stare like deer in headlights. Again I'd have to see it in color but I believe you should leave ample space between house and cover to install AND MAINTAIN gutters on house. Since they had to use 2x as long of joists to run back on roof you can probably salvage quite a bit of material for use on new cover. I leave it to you for questions.

You didn't ask but for POSSABLE property tax reasons I advise pictures like you posted here plus after pictures clearly showing new cover not touching house. See how tax assessor rated the cover and look at various ways (names assigned to structures such as this). When applying for permit ,use a structor which carries low per sq ft value. When next tax valuations come out,protest and prove your case with permit and pictures. It's vey possible cover was never added to taxes so don't wake sleeping dogs by applying for permit and/or protesting.

@klem1 thanks for your comment and solid advice. I appreciate it so much!!

I realize it's a bitter pill but I see no shortcuts. If you are covered for storm damage some hail would be a blessing right now.

The current structure could be saved and made watertight but it’s not internetable. Someone will need to pull down the drywall to inspect all the framing , fix the roof slope, re do the whole roof…