How to change an objects colour while retaining its soft edges?

The paintbucket tool is one of those tools that were originally conceived just to fill out the toolbar. It's just a magic wand followed by an automatic fill with foreground color. You don't need a dedicated tool for that.

But like everything in Photoshop - once there, it never goes away.

You need to think in more flexible terms. Select then fill is what you need here. For the first part there are zillions of options available that are much better than the magic wand.

You should be able to lock the layer opacity

and fill with the foreground using Alt backspace.  It looks like you used the paint bucket tool without anti aliasing, or an inappropriate Tolerance setting in the options bar.  The paint bucket tends to be of limited use once you get fully into Photoshop.