I can't seem to find a manual for this game, but, having just finished the tutorial mission, I don't get the impression that heroes/units carry over between missions at all. Is this actually the case? And if so, is there anything I can do to alleviate having to grind-up new heroes when starting a new shard?

Nope every shard is a new level on its own.

I was going to express concerns about having to grind, since your heroes start over, but it looks like you gain a decent amount of experience just from exploring, and sitting around grinding mostly just let's the enemy AI come in and start wrecking things. So grinding probably isn't going to be as much of an issue as in, say, Disciples?