I finally settled on a method of handling animations: An AnimationNodeBlendTree with a bunch of Blend2's and Transition's to handle switching between each of the animations smoothly. The desired velocity and actual player-state switching is handled by a custom StateMachine I implemented. The latter will need a lot more tweaking, but I think the ANBlendTree will stick around for the final product. I just need to figure out a better running-to-sliding transition (probably some kind of one-shot animation).

Here's a gif with all the animations:

The other big news is that you can now access the latest build online without having to download anything! I'm not sure yet if I'm going to continue uploading the downloadable versions. That will depend on the enthusiasm for the online version, and how much I want to keep historical copies.

You can try the web version here