Shavings let them 'burrow in' a little more closely than straw does (if we're talking regular straw here, not chopped straw) but I would not expect it to be a big difference in their warmth. Personally I find shavings to be a lot easier to keep reasonably clean, though, because they don't glump together and can very easily be stirred around, unlike whole straw.
Also, drainage in my coop is somewhat terrible after heavy rainstorms
Big puddles all over the place which I continually have to drain during the rainy season. Straw seems to soak up a bit of the water, but do pine shavings work better?
If you have poor drainage (this is in your run, right, not in the indoor coop?) then the main thing is to solve the water input problem. Make sure that roofs (including the coop, yes even if it is small) have gutters and downspouts that direct the water well away from the run; and dig a trench/ditch/swale type thing around the area to divert water elsewhere. If the chickens have compacted/eroded the ground down to where the run has areas below the level of the surrounding ground, you need to add fill to raise it up to (or above) grade.
I would absolutely not recommend putting shavings outdoors, they will quickly mix with mud and make even suckier smellier longer-lasting mud. Straw is not so great either, although better than nothing... but you have to remove it beforre it starts to rot down or mix with the mud. Best thing is to add sand or gravel when the ground is bone dry. As a temporary measure if necessary you can add coarse organic stuff like mulch or coarse tree chippings to a muddy run, but they will have to be removed before they start to rot down or mix in.
If you *are* talking about water in the coop, you REALLY REALLY need to solve that problem (intercept it, improve drainage of the whole area, and/or raise the floor of the coop). But in the meantime, the best thing for absorbing floodwaters is shavings... you let them soak up the water and then rake them out right away. Don't let damp shavings (or any other kind of bedding either) sit in the coop, that is just asking for problems, both with chicken health and coop structure.
Good luck, hope your weather improves,
Pat