Hi, your problem sounds very familiar! I had the same trouble with my long-haired rescue mog Poppy. Hopefully this might help!
When we adopted her from Cats Protection she was a tiny little thing, 2 years old but the size of kitten still. Her CP fosterer warned us that Poppy had a tendency to "fall" back in her litter tray and end up with a messy bottom, but in the first few weeks of having her we realised it wasn't her litter tray skills that were the problem. She just never bothered to groom her butt area, even though (at the time) she was less than 3.5 kg and the rest of her was impeccably tidy. Over the course of her first year with us she went on to gain about 1 kg in weight (partly her inactivity and partly us being overzealous with the Dreamies) and the problem steadily got worse as she got bigger. At its worst, we were washing her back end once a week with pet shampoo and shaving the area every fortnight. We were living in the centre of a city at the time, and Poppy was an indoor cat by her own choosing. Our garden was paved and claimed by about 5 other neighbouring cats so I can see why she wasn't keen on going out.
Now we live in a village in a rural part of the UK and Poppy regularly goes out, and has so far lost half a kg. She's still got a bit to go but the outdoor access has proved to be a blessing and I'm sure she's a happier cat because of it. Although we've always played with her - she loves Da Bird - I think the exercise from this game has only really had an effect now that she is more active.
So that's the first thing to consider - Sadie will benefit from losing the extra pounds. I would recommend taking her back to the vet and getting them to give you an estimate of what weight she should be. I don't know much about individual breeds but as she's mixed there is no hard and fast rule about what weight she should be, rather than if she were a pure bred Maine Coon in which case I imagine there are fixed guidelines for the breed.
You don't mention if Sadie is an indoor or outdoor cat, but if up until now she's been kept indoors, maybe consider giving her access to an area outside, if that's possible. If that's not possible then you'll need to reduce her food gradually, and cut out treats altogether. When we were living in the city and Poppy was at her largest, we were feeding her a whole pouch of Whiskas and unlimited biscuits every day, plus a good handful of treats. She now gets half a pouch of a higher quality wet food (70%+ meat content) a day and a very small amount of biscuits, three times a day. By small I mean about 6 or 7 individual biscuits! Believe me when I say this has solved the problem entirely, she now has a clean bottom and we trim her once a month with no need for shampoo. She is less sluggish and has a lot more energy than she used to.
If you have a browse in your local pet store for a low calorie, poop-reducing kibble, this should firm up her stools and reduce the mess left on her behind. It would also be worth feeding her a high quality, high meat content wet food as well. Mass-produced pet food is full of grains and filler and this creates smelly, enormous, messy poop (in my experience!) and is best avoided, even if it is cheap.
I hope that this is helpful, and best of luck to you and Sadie
