1. What do I need to do to assess the full extent of damage to the speakers?
Most likely the voice coil got hot enough to deform or burned enough insulation to get stuck in place, which is why it makes only soft sound, but not hot enough to completely burn out, in which case it would be dead silent. To assess this remove the tweeter and hook it up directly to the amp. Full range will not damage a tweeter, if the volume is low, so if it is still good you will know, it will sound like its getting a full range signal. In that case its crossover damage. Which is unlikely. Almost certainly the tweeter. But you asked how to assess. This is how you assess.
The tweeter will probably be marked with a part number or something. Do a search, locate the replacement, swap em out. No big deal.
Given how new it is and what happened you might also try calling the manufacturer. Honestly explain exactly what happened, throw yourself on the mercy of the court. They may offer to send you a replacement. Or let you buy one from them. At the very least they will probably help you find the correct replacement.
You don’t mention the speaker, or anything. Just so you know, a lot of the higher end speakers use matched drivers. So when you call be prepared with serial numbers, parts numbers, purchase details, etc. Manufacturers have a lot of clueless customers. They make their policies accordingly. That’s why more often than not they require you ship the whole thing back. Like you can’t turn a screw-driver. Because frankly, many cannot. The more you appear to know what you’re doing the more likely they are to send a replacement, which would be my Plan A.
There are no dealers nearby. 2. How can I prevent this from happening again?
Only way to truly prevent this is to fuse the speaker or tweeter. But I don’t recommend it. Much better to learn your lesson: avoid digital. Nothing good ever comes of it.