Your hot shut off valve seems to be a multi-turn valve, which would have an internal rubber washer on the end of the shaft to fully seal the opening when the valve is shut off. Especially on a hot water line, this rubber washer degrades over time and no longer fully seals when the valve is off.
Side view of typical multi turn shutoff valve:
Fortunately, it's pretty easy to replace the washer (and the washer sealing around the shaft where it exits to meet the handle) without removing the valve. Just shut off water upstream, release pressure, and carefully undo the packing nut (the nut just under the handle) to remove the whole shaft assembly. Either replace the two washers or the whole shaft and reassemble, adjusting the packing nut tightness to prevent leaking around the shaft.
Here's a photo of a typical replacement kit:
You're looking for an "angle stop rebuild kit". I would purchase one with a proper brass metal shaft rather than the plastic one in the photo. Any plumbing supply shop online or brick n mortar should be able to get you the right kit.
Ask This Old House has a YouTube video that shows how to do the rebuild.