I'm running into an issue where I can use any number of Hit Dice and it will not modify my HP at all. Long rests work correctly and restore me back to max HP and reset my hit dice.

I've seen some mention that other abilities may not be modified by short rests as they're supposed to, but I haven't confirmed this myself.

DDB doesn't roll dice. You are expected to roll the hit dice and add HP yourself. The hit dice check boxes are just for keeping track.

Ahhh...yeah I knew in general about DDB not rolling dice, I just hadn't applied that knowledge to short rests. With long rests being an "automatic" feature, I just had this idea that short rests would be as well.

My ignorance = humiliation

Are there plans for DDB to eventually roll dice?

You can see all planned features and their priority on the roadmap: https://ddb.ac/feature-roadmap

It is a long term goal, will involve multiple teams, and is best added after or alongside other things on the list. I would expect it around the same time as a VTT.