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You can try Spatial Join tool (Analysis toolbox - Overlay toolset) with the INTERSECT match option.
Thank you, Dan! That almost worked. My .shp files seem to be in different spatial coordinates (one is GSC_North_American_1983, the other is GSC_WGS_1984). I tried a few transformations, but there doesn't appear to be an exact one. As a result, some counties get matched to multiple divisions (I allowed, "one-to-many" match) because their borders spill over a tiny bit. I've tried "one-to-one" matching and that seems to work, but I'm not sure how it chooses which division to match to - would it be the one that matches best (that would be ideal)?
The inconsistency in boundaries from different data sources may exist, even when the datasets are in the same coordinate system. So, instead of using the county polygons, you can try to use points that represent the polygons:
1. Use Feature To Point tool (Data Management toolbox - Features toolset) with the "Inside" option to get county_points from county_polygons. Each point carrries the county_polygons FID and attributes.
2. Use Spatial Join tool to find INTERSECT matches between county_points and the division_polygons. Now the spatialJoined_county_points have division polygon attributes.
3. Use Join Field to join county_polygons with the spatialJoined_county_points via the county_polygons FID to transfer the division attributes to county_polygons.
That works perfectly, thank you!
You are welcome! I am glad it worked for you.