i tried to put the catalog and photos in my Dropbox. The photos obviously transfer and the .lrdata file transfers but the .lrcat does not.
It should transfer as long as you remember to quit Lightroom every single time before moving to the other computer. If you don't quit, there is a MAJOR probability of corruption of the catalog which is why you have to be EXTREMELY careful if you put your catalog in a dropbox folder. It works but is quite dangerous.
You also have to allow time for the catalog to upload on computer 1 before you shut it down, and time for the catalog to download before you open it on computer 2. Check that the Dropbox icon says Up to date.