I would say that straw/sawdust is not an either-or issue. They do different jobs in a cob mix.
Straw (or other fibre) is for tensile strength and I would say is necessary in any cob wall.
Sawdust (or other lightweight additive) doesn't add tensile strength but improves insulation value and reduces weight, possibly at a cost in compressive strength. OTOH if you have less weight, you need less compressive strength, as much of the weight the wall needs to hold up is the wall itself.
In the wall of our cob house
, we replaced half the sand with wood shavings (basically coarse sawdust) to improve insulation value. So the mix by volume was 40% clay soil, 30% sand and 30% wood shavings, plus straw -- compared with a normal mix of 40% clay soil and 60% sand that we used in the cabin
we built first.
I did some basic compressive strength tests (measuring the load required to crush various bricks of each type) which indicated that the compressive strength of the sand-sawdust-clay mix was not significantly different to the normal sand-clay mix. (See photo of compressive strength test. Please note the highly sophisticated and expensive scientific equipment.)
This result was good enough for me, so we built the main wall using this mix. It's held up extremely well, is definitely warmer than standard cob, and has an added minor advantage over the standard mix, that you can drill or nail into it. Only disadvantages are that it dries more slowly and shrinks more when drying than a standard cob mix.