Specifically, the first flight for my honeymoon just got rescheduled to 12 hours prior, during my wedding. The airline says they have no flights on them or their partner airlines that get me to my destination on time. They have offered a full refund, but the possible replacement flights are all much more expensive than that. What can/should I do?
That sucks, but unfortunately all the airline is required to do is do their best to find alternate transport or offer the full refund, the 'contract of carriage' for the airline will spell this out (although they're sometimes difficult to find). You don't say who you're flying but you're in the US so I'll pick United as an example, others will be similar, from here (PDF):
Rule 24 ...
(Rule 27A is about the technicalities of refunds).
Now I'm assuming this is a probably because a) it's your honeymoon and b) you've probably spent a money on the resort, possibly arranged the time off, etc. You also say it's your 'first flight' so I assume there are others, it's not clear if they were part of the refund.
The first thing is to check any applicable travel insurance, if you booked on credit cards or through an agent you may have some. This may refund some or all of the other out of pocket costs (alternatively they may pay for the more expensive airfare). Find out what, if any insurance applies to you and call them and explain.
I would also make it clear to the airline that it's your honeymoon, you have a lot of emotional and financial investment in it and ask, nicely, if there's any way some other solution could be found. Maybe try and get hold of a reasonably high up manager, or maybe someone in corporate PR. A PR person may see a potential good PR story (or the change to avoid a bad one) and be able to pull more strings.
Although, just to be clear again, they're not in the wrong here -- although the fact that schedules are not guaranteed is not a fact they tend to advertise ...
If that fails I see two alternate options -- you could try your luck. Ask the airline to get you on standby with the most suitable alternate flights (several if possible) and hope that someone no-shows and you get the seats. At the same time keep looking for other cheaper flights.
Alternatively ask where the airline can fly you to on that day, and see if you can work it out from there. This may mean a convoluted route, or even going somewhere else entirely.
Without knowing the airline and your eventually destination I can't really suggest anything else. Good luck with it.
Some airlines are quite keen on having a good social media appearance. You could go for a polite online naming & shaming approach (e.g. Facebook, Twitter, Blog, etc). All you need to do is to include the facebook or twitter account of the airline in your post.
As @SpaceDog says, the regulations in place are well taken care of. That doesn't mean the airlines don't have the power to step out of the ordinary, and offer you an alternative. So if you keep it civil and write a sad blog on how your honeymoon got wrecked by the airline, you might have a (very small) chance that they'll do something.
This approach did work for me on various occasions. Once ground staff threw the book on me because of some regulation and denied me boarding. They didn't even want to listen to my explanation why the given rules didn't apply. Complaining about it on twitter solved it in 5 minutes. "Please, go to desk XX, where you will be given a new boarding card".
BTW this approach only works if you keep a subtle balance between complaints and compliments, and if your online affiliates are potential customers for that specific airline. Having a facebook or twitter account with only a handful of followers doesn't help.