I would definitely recommend investing in an air purifier, I have 3 kitties of my own and it helps a lot with dander and any other pollutants you have affecting your health.

I always recommend the Pink Stuff cleaner, I prefer the spray bottle but if you have some stuff to scrub at, the paste is very helpful.

To not overwhelm yourself, try focusing on zones. Prioritize the most frequently used spaces in your home and take them down one by one, and sanitize as you clear things out.

Breaking old habits can be very challenging and rewarding yourself along the way may keep you from losing motivation. Good luck to you both!