You can still overstuff your barns. Just tried, and it worked without a problem - by Judy Cowan
1  First, make sure you have 200 animals in your barn. 
2 Click on your barn and click on view animals. When this opens, you'll be given an option to "move" your animals. 
3 Click move, and your animals will be put into storage. This is the only time you will have to do this. 
4 Next, enter offline mode, then right click on Play and open the game in a new tab. Once the tab is open, click to go offline again, and right click play and open game in a new tab ) again, and again, go offline.  Repeat this process untill you have as many windows as you want opened, (I use 10) which gives me 2000 animals.
5  Next, go back to you first farm, open up your storage, pull up your animals and click the option to move them into your barn. Repeat this in all of your open tabs. 
6 Once you have the last one finished, close all but the last tab - on the last tab, click your play button to refresh the game and return to online status. 
7 Open your barn, and you should see that is overstuffed. I just tried this with my rabbits on one of my small farms, and it worked fine. Happy Farming Everyone!