I'm not sure what type of wood your are cutting but you need to keep the blade clean. I use water and windshield washer fluid in the winter time. A dirty blade is thick and it will rub either the board or the log, or both. When it rubs it gets hot. When it gets hot it gets longer. When it gets longer you lose tension. Without a gage to look at and see that it has lost tension you may need to increase the tension once you get it running for a while.

Always back off your tension when leaving the saw, even for a short lunch break. Never leave the blade tensioned up over night.

You could/should try and see what the manufacturer of the mill recommends for band wheel belts/tires and see if they are flat or crowned. If this oscillating happens on every blade then it could be these tires/belts.

Good luck with your research and solutions.

Jim Rogers