I understand being budget conscience. But, have you priced good quality ZTRs? A new one will set you back just as much as a tractor. Sure, there are "low cost" units out there. But, with that much grass, they won't last or perform like you will need them to.
Any way you cut it (punn intended), mowing 11 acres will not come cheap.
Your best bang for the buck is to go used. There are a lot of good machines out there that would work for you and still be reliable. Something like a BX or a JD455 for the "finished" yard, and and older Ford or Deere for the other field and loader work.
Want to save even more, consider refurbished Yanmars. Small ones, that are prefect for mowing can be found really cheap.
BTW, one steep bank brand ZTR that would work for you is the Kut Kwik. But finding a used one that is still affordable might be hard. I saw one locally here about a year ago for $7K. I think it had a 7' mower on it. It was a steal at that price.
When the ground is saturated everything will tear it up, even you as you point out. I would also think that, given such conditions exist, the affect of whatever machine would be far less important to your decision processs, since they all would cause unacceptable damage. Doesn't that mean the only recourse is to stay off of it completely when it's like that?