I would have to go with a shed. We used a cover until on a nice cold winter forced some vermin to make their home under the cover. They feasted on my rubber roof, air conditioner, vents, awning and the cover itself.
Since that happened, we have had a shed at our last two homes and really like them. No new vermin problems. The damage to the rig cost almost as much as the shed and pad.
If you have enough room, I would get a shed that is at least 5' longer then your rig and wide enough to open any slides. Also, adding gable panels to the back end and at least one row of side panels to keep the sun/weather off really makes a difference. Just make sure to check the clearance height and width with the sales person to make sure your rig fits with roof slopes on the shed.
Regarding getting stuck in the yard, yes have been there too. I would suggest purchasing a roll of geotech fabric, in whatever width and length you need and placing gravel, millings, etc. on top of the fabric, maybe 3" or 4" thick. The gravel will not sink into the soil with the fabric under it. The gravel pad should also extend under the shed, so the shed installers have a good solid surface to install the bottom rails.