I actually figured this one out, what happened was: on a clip I created of my study area, I believe I let it set the nodata values as 255 (the ArcGIS default). Although for some reason this did not show up under properties when I looked afterwards. When I redid the clip and told it to assign 0 to nodata values, redid the extract by attributes tool creating a new raster for each landcover type (from this new clip), then redid the zonal statistics tool, it came out correctly with no negative values. So, what was happening was, since numbers were assigned for each landcover type (for ex: water =11) the sum was calculated by 11+11+11+11... and so on. However, with nodata pixels set to 255, cells that were not water were calculated as 255. So the tool was doing 11-255+11-255... By redoing all those steps and changing the nodata pixels to 0 in the original clip, the tool calculated sum by doing 11-0+11-0 .... I still don't know why it only did the 11-255+11-255 thing for some zones and not others, but my fix seems to have worked for all zones.
Also, to answer your question, I wanted to see if I could make this tool work rather than performing a join with the vector layer because it allows me to show which pixels (that appear split by a county line) were counted in which county zone. Also keeping it in raster rather than vector will become helpful later in my analysis I believe.
HOWEVER, now I am running into a different issue. After I fixed the problem above, I started running the zonal statistics tool on each extracted landcover type raster (so one for the water raster, one for the deciduous forest raster and etc). It worked perfectly on the first 4 landcover types. Then, without changing anything, when I was on the 5th landcover type, the zonal statistics tool crashed and gave me the generic 999999 Error. It also gave me an error that said "no current record", and I am not sure what that means exactly. I am particularly confused because I changed nothing between when it worked and when it crashed. I was just sitting next to the computer taking notes on paper, then each time the tool finished I would just set it up and press go for the next landcover type.
So, to troubleshoot I tried it on a landcover raster that the tool had previously worked on moments before (exact same input raster, zones and everything) and the tool crashed and gave me the same errors. So I am completely confused why the tool worked one moment and crashed the next, even when I tried to re-run it on an input raster that worked less than an hour before. Do you have any idea why this would happen?
Thank you for all your help, I really appreciate it.