My timber retaining wall is tipping over
It's quite hard to tell how big this wall is but you stated it's 3-4' tall.

It looks like it was built simply as a vertical wall. There's nothing you can do about it. It was inevitably going to tip over.  So yes, you do have to dig it out.

If you want to replace it with timber, you need to add something called deadmen to it. A deadman is a timber that runs perpendicular to the wall itself that is then attached to another timber that runs parallel to the wall itself. Diagram:



Without those, your timber wall will always end up getting pushed over eventually. 

The catch is, of course, to bury the deadman, you need to excavate all the dirt behind the wall back to about the same depth as the wall...so you'd be moving the 4' of dirt behind the wall as well. That may or may not work depending on who's side of the yard that is. 

Another option is to go with a block wall. Retainer block walls are typically staggered backwards as it goes up. You then line the back side with drainable material (river rock) and a drain tile to allow any water building up a route of escape. However you're pushing the limits of most consumer-size retaining wall blocks at 4'. 

Yet another option is to build two, 2' walls stepped back (one 2' wall on the bottom, then a second 2' wall several feet behind and above the first one). This is likely the best solution for long term retaining of the land. Plus it may offer some landscaping options (allow you to plant the strip in between with decorative plantings).