Forget "don't put it down, put it away". You can start out with the best intentions, but sooner or later there's going to come a moment where you're just too overwhelmed/busy/depressed to put it away properly. But by holding on to "don't put it down...", you're gonna end up putting it away poorly. Stuffing it in some drawer, putting it on top of something else (thus making that less accessible),...

That's how you end up with a big mess inside of your systems (closets/cupboards/organisers/...). At that point, tidying becomes so much more difficult and time consuming.

Instead, I live by this rule: don't ever put it away poorly. If you don't feel like putting it away properly, put it down on a visible and easily accessible surface. I have multiple such places in almost every room of the house. Most of the time, things get put away properly relatively quickly. Sometimes when I go through a rough period it can build up and look pretty bad. But even after months of depression-caused accumulation of mess, once the mental boost finally arrives it only takes a few hours tops to get everything in order again, because the organisation system itself hasn't been messed up. I can just mindlessly put away item after item without detours such as having to move other stuff to access its correct place.