Just wanted to add here for the benefit of others, that after further testing I found that changing the permissions didn't actually do the trick at all. I thought it did because I tried one particular block, which also had another setting changed, thinking it was the permission change that did it. When I accidentally tried the other HTML blocks they were still editable.
The actual setting that did the trick was the following:
1) Go to the Default Dashboard
2) Configure each default block and for "Where this block appears", set "Select pages" to "Any page matching the above".
What this magic setting appears to do is treat the blocks now as a single instance, as oppose to each user having their own instance on their dashboard. Now if you try to access your dashboard as a non-admin and click configure, you cannot change any of the settings including the HTML. Only an admin can make these changes and they will update for everyone across the platform.
Hopefully this isn't just a glitch that will be bug fixed later, because it works perfectly for this use case.