I have about four on each boat. My spare ones are the floating kind...they aren't great winch handles, in my opinion, but I'll be damned if I'll lose the last ones!
My habit, such as it is, is to leave only one winch handle in use in the weather-side winch. I will keep a spare in a pocket or otherwise secured in the cockpit. I also prefer to raise halyards from the mast and will keep a winch handle in a pocket there.
As part of tacking with a crew, I will ask for the winch handle to be handed up to "the high side", and I make sure the winch is still "pre-loaded" with two to four wraps (depending on the winds and the ultimate point of sail). Then I say "ready about", put the helm down and release the now lee sheet just as the sail is crossing the centerline. By this point, the crew is grinding and I start to haul in. This can mean I am handling both sheets at once: letting the lee sheet run out in a controlled fashion as I am hauling in the weather sheet. I will haul in the weather sheet one-handed for the first little bit and then two-handed as needed. Obviously, this works best on a boat with a smaller cockpit and a tiller, as I can steer with my knees!
My reason for not just throwing off the lee sheet is speed and the reduction of wear on the foresail due to flogging...I want the wind and not the sheets to move the sail to the new side, and if I essentially keep a bit of tension on the clew as I tack, the sail doesn't flog, and the likelihood of needing to skirt a big genoa is, I find, reduced. Anyway, this works for me.
Even though we cruise, we like to handle sails and tack and gybe as if we were racing, primarily because we like to sail efficiently, and also because so much of the boat's power is based in inertia of all that keel weigh moving forward. If you can efficiently tack and gybe, you keep boat speed up and can keep the sails drawing. This is also why I tend to "over-tack" slightly, to get the sails pulling nicely and steering to a close-hauled angle as we are grinding the sheet in. All these methods I learned from racing, although in club racing, it is rare to see them all on the same boat, or being done smoothly. Backwinding the genoa or jib slightly mid-tack is something I only see on smaller boats, (and it doesn't work that well in light airs), but it's a good method. As Alex is often saying in his videos: try it, experiment, see if you like it.
About the only thing I would change in my current set-up is to switch to those one-handed locking winch handles. My wife has smallish hands and my son is seven: being able to remove a winch handle one-handed as I can usually do would be a big advantage to them because they could keep one hand free for the boat or to better hand the tailer the sheet for a nice, fast sheeting in.