I would advise you against saw
dust as it's very fine and can cause respiratory problems and ear/eye problems if it gets into those.
I use shavings in the litter tray as it absorbs wee quite quickly, helping to avoid them getting wet feet.
In my rabbits' bed chamber in the hutch I have 2 decent newspapers i.e. the Sun on Saturday as it's quite thick, then a piece of old carpet cut to size and on top of that I have hay that they wouldn't eat. However, once the temperature starts to drop I shall replace that hay with straw as straw is warmer than hay. Yes it has less nutritional value if they eat it, but it holds the heat in better.
We get Carefresh for the hamster. I wouldn't get it for the rabbits as its too expensive considering the amount you'd need, but that's my opinion