You can still go full Attack on Stan, but it is much harder (and risky). Balancing Health, Attack, and Defense is key now. Because the Stamina mechanic has been removed, being tanky enough to live for 10 to 15 turns is optimal for winning the most battles.
The last update was about 20 hours ago, so I don't think that one was because of an update. Some people never have issues with the saving on the web version. Some people are constantly having issues with the saving on the web version. I don't think this is an issue I can reliably fix until Cloud Saving is the main saving method. The only thing I can suggest is downloading the game, as that won't have any save issues.