Adjust the hitch.

Most Goosenecks hitches can be adjusted up & down. On my F350 I had to raise my hitch one hole (about 2") higher to clear my bed rails (I've got a roll up bed cover and the rails stick up another inch higher than the stock truck sides).  You don't want to go too high or all the weight  ends up on the rear axle and you drag your trailer bumper when you cross streams or dips.

If you have to raise the gooseneck hitch so high to clear the bed that it puts the trailer on an angle, you may need to put a lift on the axles and raise the overall trailer height. I see this frequently any way. As folks try to increase the ground clearance for back country or forest service roads.  It's another 2" or so of step up for the horse, but at the same time adds two inches of ground clearance at the rear bumper, so you don't drag. It's not something you see a lot at the fairground or rodeo grounds, But ranchers, outfitters and endurance riders do it a lot.

and a couple whip antennas.

Back to your question...we had the same problem with an older 4h hooked to a '04 Dodge.  The drop wall is too short.  We could raise the neck/coupler but all that did was raise the front of the trailer - not off the bed.
Had to get a new trailer - hated that. 

I have a lifted '03 chev 2500hd diesel and bought a '93 sooner GN. We had major clearance problems. The trailer has dexter tortional axles so I cut the entire axle assembly loose and welded in 3 1/2" box tubing between the axle & frame. Was a pretty simple fix and have talked to many dealers that will do it for you.

Good Luck, CS