Save the tuber, its fine as long as it doesn't start to rot. It would be best to cut that broken piece off of the main tuber and give yourself a clean cut. You can then cover that with any liquid fertilizer, not to make it grow but just to keep it clean and free from anything that might attack the open tuber flesh. Dahlias bloom from the crown, or the part of the tuber that connected it to the main bulk of the tuber cluster. Depending on the size of that connection you may get 4-8 blooms out of a crown. The end that is broken is not that crown end, so you've lost nothing there.

Depending on your weather, you can plant that tuber cut end down, leaving the crown exposed, and watch it develop. As it does you can add soil around the crown to build up the soil that will support the stems.

The small tubers should grow just fine. As long as the crown is intact. The crown can often get ruined when it is cut from the tuber cluster, so for small ones I either leave them attached to the cluster, or cut of a substantial piece of the tuber they are connected to so the crown has lots of material to grow from.

Best of luck.

Cheers

Russ