Last year I moved from San Diego to Portland, ~1100 miles. This included moving my entire brewery, and lots of beer (full bottles, kegs, and yes, even full carboys). We ended up going with PODs, which means that we packed and loaded everything ourselves. Everything arrived completely unscathed. It's really not all that difficult to pack this stuff. Bottles went into normal cases, after all, that's how the distributors move beer. Empty carboys were wrapped in towels and placed in a secure spot where they wound't move. Full carboys were placed into empty buckets, carboy boxes, or milk crates, and packed with towels or other padding. Basically as long as none of this stuff is dropped, it should be fine.
I know you say you're busy, but take the extra time to pack this stuff yourself, and safely. Don't rely on the moving company to pack specialized items (brewery stuff, or otherwise). I'm sure you'll be driving to you new place more than a single time, and it's really not too difficult to place your more fragile stuff in the back seat of your car when you do. Hardly takes any time and gives you the extra piece of mind.