Excellent feedback, thank you! Good observations, the back patio does slope a bit down toward the run which I'm sure is not helping.
What about a solution like this one? I could put it up on the patio. This should be sufficient indoor space for them and easy access for me. Do you think I could use the same run but move it up on the slab as well or would I still need to increase the size of it?
Considering the frustrations I have with my current setup and my non-handyman status I'm not opposed to selling it and upgrading.
Personally, I would rather see you address your birds' need for more run space before you increase their coop space. My own birds spend very little time in the coop, preferring instead to be out in the covered run all day. We were in the middle of a blizzard last winter and my birds were still out in the run. They pretty much just use the coop for laying eggs and roosting at night. In general, I think a chicken keeper can get away with a smaller than ideal coop if they have a decent sized, covered run the birds can access in all weather.
Yes, it would be nice if you got that bigger coop, but your birds will still want to spend their time in the run and so you'll still have the same problem with poop and wet that you have now. I would put priority on getting a bigger, covered run. As someone else suggested, a covered dog kennel can be an inexpensive option for a "not-so-handy" person to provide a dry run and a decent amount of predator protection.