I was in a similar situation and my local shop wanted $150 per carb for cleaning, I'm pretty mechanically inclined and decided to do it myself and I'm glad I did, it's not nearly as difficult as you might think

 

Well I'm reasonably convinced to do it myself. I'm still going to call around but I don't doubt I could do it.

Would carbs that need rebuilding cause fuel in the crankcase though? I feel like the problem may be more serious than just the carbs.

 

Fuel in the oil is a normal symptom of faulty carbs.

 

So here is an update. I decided not to have the Honda shop mess with anything. They hauled it back home for me. I took apart the carbs and cleaned all the jets, checked the diaphragms etc. From my untrained eye, I didn't see anything amiss. The carbs were already pretty clean. I did notice an oily fluid around the carb stacks and inside the air box. I'm guessing that would be from the PCV-type connection to the air box. I also drained the oil, which was mostly gas.

I have a feeling by just cleaning the carbs, it may not fix the issue. Like I said, they already seemed pretty clean. Jets weren't clogged and air blew through the lines fine.

If once I put it all back together, and it still doesn't run right, I'm not sure where to go from here.