Engaging the auger is hardly load. That is just parasitic loss. The OP said it runs fine once the auger is engaged so there is nothing to adjust on the fuel side. The stutter is very common with Tecumseh Snow Kings of all sizes. I've heard hundreds of them doing that to some degree over the years of snow removal as well as purchasing and selling used models. Spending time and/or money on this is a waste. If it is losing power and sputtering while removing snow, then it needs to be addressed. I've wasted time and money tearing down my 8.5hp to adjust the valves and it did nothing to minimize that very same sputter at idle.
reynoldston is correct and it's exactly what I've done for years. Ignore it and remove snow.