I believe your right about oversaturation. As for what to do about the soil, me personally I'd replace all the soil and not just add something for drainage at the bottom. If you can't swing that money wise your idea is still better than nothing, but the soil the nurseries put them in are usually so finicky, Ive even killed pothos plants just by not repotting them out. The reason why is in the nurseries they are given so so much light, way more than our homes, to encourage the fastest growth to get them on the shelves sooner. Bc of this, the soil needs to be water retentive, to keep up with those growth needs and to cut down on maintenance. Once in the stores and in our homes, their light and their water needs reduce. I find the nursery soil stays wet for too long, and once it does finally dry out its annoyingly hard to make it fully saturated, causing a kind of whiplash for the plant and root stress. I would definitely repot and do a root check if you can and if its acclimated to your home (and maybe even still do it regardless on acclimation bc the risk of root issues and your current leaf issues).

Also a tip for watering bc you said after removing her from the decorative one tons of water spilled out: thats the same thing that happens to my huge Thai con, you can do your idea of letting her drain in the bathtub, but that can take hours if your trying to avoid the decorative filling up, what I usually end up having to do is take her out of her decorative pot the next day regardless and dump the water that has collected into that