I’ve combined a bunch of suggestions/ideas, and this is my current favorite process for cutting paper/cardstock.
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Low and slow: speed 200, at whatever power it takes to cut. For regular paper, that’s something around 2. For the cardstock I have, 15-20.
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Medium tack front masking. I ended up with this one and not the transfer paper @jules recommended. I wasn’t going to buy any at all, except I needed a kicker to get me over the prime | FREE One-Day limit on a different item. Works pretty well, but you need to add a little bit of power when using it. Very easy to apply and remove. Kind of expensive in the 30’ length, though; I calculate it to be about $.50/sheet of US Letter paper, so if I find myself using it regularly, I will get the 100’ rolls.
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Krylon Easy-Tack Spray Adhesive (or similar) and some plywood. I cut a piece of Home Depot plywood to 9.5x12", to leave a half inch border on letter paper. Then I sprayed some Krylon adhesive on it. This is what Jules recommended to me a long time ago to retackify a Silhouette cutting mat, and it works just as well for this. The plywood acts as a reusable cutting mat, and as a bonus, keeps bits from flying off as confetti and potentially interfering with cuts.
This is a finished test cut:
And the result:
With the right settings, weeding is more like “poking out”.