jgall19 Offline mode blocks Gallery access, and it will also block you from starting the limited-time events or redeeming your progress. However, you can occasionally go online to get what you want and then go back offline afterwards. For example, the next event starts tomorrow, so you could go online once, start the event, go back offline, and only go online again (briefly) when you've completed all the tasks in-game and you want that progress credited to your account. You only need to be online once (per set of tasks) to show EA servers you've finished the tasks, and then the rewards are yours permanently going forward.
A less drastic alternative you can try is keeping your computer online but placing the EA App in offline mode, or even leaving the App online but disabling Online Social Features within Sims 4, under Options > Game Options > Other. It's not always clear which setting is the issue. Either of these alternatives would still block Gallery access and the special events, but the above method of occasionally restoring the connection would work equally well.