Years ago I had an F2 Gleaner.  It was a 2 wheel drive with 23.1x26 rice tires on it.  It was light and well balanced ......everything pretty much centered on the combine .....cab, engine, grain hopper giving good weight distribution on the tires.  Went good in the mud.  Heads now are cheap for them....can get the newer black series head or older orange heads.  Are easy to work on. One of them would fit in your budget. 

 As others have said a 1440 International would also work but the combine/corn head is going to cost more than the F2.  Of course you will be getting more combine for the money and will have more capacity and be easier on the operator.   I now own a 1460 and the difference between it and the F2 are significant but would still run an F2 if circumstances dictated.

Have never owned a Deere so have no comment on them other than I am sure they have a good combine that would fit your needs.  

Think I would just start doing an intense search for combines in your price range and see if I could find a machine that was relatively low hour, been shedded and well maintained.   Would not be so much focused on brand as I would condition and price.  

One other option I just thought of....offer a premium to someone to custom harvest your corn.  Throwing a few extra dollars out there might get a little better response and wind up costing little money vs leaving the crop in the field longer.  

Good luck whatever you decide to do.