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 ...
3) Schedules are Subject To Change Without Notice - Times shown on ticket, timetable or elsewhere are not guaranteed and form no part of this contract. UA will promptly provide Passengers the best available information regarding known delays, cancellations, misconnections and diversions, but UA is not liable for any misstatements or other errors or omissions in connection with providing such information. No employee, agent or representative of UA can bind UA legally by reason of any statements relating to flight status or other information. Except to the extent provided in this Rule, UA shall not be liable for failing to operate any flight according to schedule, or for any change in flight schedule, with or without notice to the passenger.
C) Change in Schedule - When a Passenger’s Ticketed flight is affected because of a Change in Schedule, UA will, at its election, arrange one of the following:
1) Transport the Passenger on its own flights, subject to availability, to the Destination, next Stopover point, or transfer point shown on its portion of the Ticket, without Stopover in the same class of service, at no additional cost to the Passenger;
2) At UA’s discretion, reroute Passengers over the lines of one or more carriers in an equivalent class of service when a Change in Schedule results in the cancellation of all UA service between two cities;
3) Advise the Passenger that the value of his or her Ticket may be applied toward future travel on United within one year from the date of issue without a change or reissue fee; or
4) If the Passenger is not transported as provided in C) 1) or 2) above and does not choose to apply the value of his or her Ticket toward future travel as provided in C) 3) above, the Passenger will be eligible for a refund upon request. See Rule 27 A).
(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.