Tiagra levers moving loose

I'm not sure why you've not received an answer before now. The fist thing I would do is change the cables and housings. This is cheap and easy. You can get a cable kit on ebay for about $15, and you can do it without any special tools*

It solves so many shifting problems because friction builds up in the housings after use and time.

It is not a guaranteed fix, but it is something you want to do periodically anyway. There are a jillion youtube videos explaining how to change cables and housings, but let me know if you want me to recommend one in particular.

*I mention no special tools. Technically, a cable cutter is your best bet for cutting the cables, but not practical if you don't expect to be doing this much. If you decide to hack it with a pair nippers of some other kind, you'll mash the housing. Just make sure you try to return it to round as best you can with a pair of pliers. Then take a toothpick or pencil or similar to make sure the hole (for the cable) is not obstructed by any of the metal that you mashed. This is easier with shifter housing than brake housing.
I have had a lot of success with a pair of dual action mini bolt cutters.

When you take the old housing off, use it to set the length of the new housing that you will cut.

Note that shifter housing and brake housing are not the same, nor are the cables. Shifter housing is slightly narrower, and has wires in a gently spiral running almost parallel to the housing. Brake cable has a tight spiral and running almost perpendicular to the housing. Shifter cable is 4mm, and the cable for it is 1.2mm.

I'd suspect the cold has gummed up the notches into which the pawls rest, inside the shifter's mechanism.

This is a fairly common problem on older shifters, where you can swing the change lever to its limits and nothing catches so no shift happens.

A quick test is to flood the inside of the shifter with a solvent, like WD40 and then work the mechanism to both extremes. If this fixes the problem, you're all good. If the problem returns fairly quickly then you'll need to do more of a teardown and deep clean on that shifter.

I've got a 105 right-hand shifter that does this on cold days. I've not had time to get it all apart, so a monthly shot of CRC-556 sorts it for a while. Not ideal but better than being unable to change gears. And its not lubricated very well because of this.

Cold makes this worse because the grease inside gets stiffer, and won't get pushed out of the way as easily as on a warm day. So the leading edge of the pawl doesn't click into the cog's tooth like it should - there's a wedge of harder grease in the way.