Can you “backwash” a sawyer filter with a bit of air to try and dry it out a little bit to help prevent freezing or is that going to ruin it

What about putting the caps on and keeping in your pocket where your ambient body heat can keep it from freezing?

I do this when packing up camp to avoid carrying extra water weight. If it’s below freezing overnight, best to at least keep the filter with you in your tent overnight to prevent freezing. I’ve never been confident that I could blow out enough water to prevent ice damage.

You won't damage it, but you won't accomplish much either. Keep it in your jacket, or sleeping bag if you're sleeping, to keep it from freezing. 

Also, if you drank from the inlet end of the filter, you're drinking unfiltered water. 

A lady from Sawyer customer service told me the fibers are so small that they never really dry, even if you do push as much water out as you can. So there’s always the risk that if it’s had water in it somewhat recently, any water that’s in a pore or fiber could freeze and expand therefore diminishing the filtration capability. It’s going to be a best practice to always keep the filter in a ziploc in your sleeping bag during freezing temperatures.

I am not sure anyone here can answer that question. My guess is that it would break the filter. You only need a very small percentage of the filter media to fail to be a big problem. A small broken area due to freezing has a good chance of passing a disproportionately large amount of water (because it might have a lower resistance to flow/required head pressure). As others say, put it in your sleeping bag at night.

Do not place a water filter onto your lips and risk ingesting contaminated water. There are better ways. :)

Gravity, and inerta. Sling that sucker a few times to clear out most of the water, then drop it vertically into a pouch or pocket, with the 'clean end' facing up, and it'll basically drip dry itself before nightfall when the freezing temps hit.

Also, if you have a little zippered baggie for that thing, bring it with you into your sleeping bag at night, just like you do with the water supply you'll want to be sure is un-frozeny when you wake up in the morning.

I don't think you'd be able to get enough of the water out this way. Best to keep it in your sleeping bag next to you

Running air through a sawyer filter will not ruin it at pressures you're likely to get without a compressor, so that's not going to ruin it. Sawyer recommends you treat a filter as destroyed if it has been frozen any time after the first time it had water in it, even if it was "dry" for a year because the tiny little tubes inside never really dry out. Worse, there's no valid way to (home or field) test the efficacy of a sawyer filter, so if you failed there's no way to know until you get sick. So allowing it to freeze after this will probably ruin it and you won't know whether it did or not.