How do I get my Envelopener to cut cleanly every time?
I use envelopeners a lot and indeed prefer them to any other means of cutting paper in almost all cases. Here are some tips from my experience:

Regarding the cutting tool itself:


First of all, envelopeners are consumables: they will become blunt over time and there's not much you can do about it. Since they are cheap, simply replace them once in a while, especially if you start running into issues more often than you used to. Your mileage may vary depending on how often and what types of paper you cut, but mine last less than half a year.
The blade which sits inside the tool is pressed between two halves of the plastic case. You can open the case by inserting a fingernail or a screwdriver blade into the narrow gap between the two halves. This will allow you to:


Clean the blade from paper residue which gathers around it. Some envelopeners have a shaft which is supposed to allow the lint to escape from the blade but this does not always work perfectly.
When closing the case, press the two pieces of the case firmly together. Sometimes the two halves become a little loose, causing the blade to not be mounted in a fully symmetric manner, which can cause both parts of the paper to move around a single side of blade instead of getting the blade exactly in between them.



As for the cutting technique:


The envelope opener works best if the crease along which you cut is perpendicular to the paper edge. So, it's good for cutting squares but much worse for cutting hexagons. Personally, I use a Swiss Army knife or scissors for cutting hexagons since the envelopener doesn't handle them well.
Sometimes you run into problems such as splitting the sheet into layers instead of cutting along the edge when trying to cut thick paper and the crease along which you cut is pressed too firmly (e.g. using a bone folder). So, sometimes it helps to either not make the crease that sharp, or to fold it in the reverse direction after folding, which also results in a somewhat less sharp fold.
As mentioned above, an envelope opener consists of two plastics casing parts with the blade fixed in the middle right between them. These two halves of the case do not always perfectly fit together and there may be a small gap between them. Sometimes when you start cutting, you may press the paper too deep inside the casing, which causes the creased part of the paper to get stuck in this narrow gap. The solution is to check that you are not inserting the paper too deep.


While these tips do not guarantee a 100% success rate, and I still occasionally make a bad cut, I think that I have become better at using the envelopener over time, so I hope they are helpful for others, too.

UPDATE: I summarized these tips and some more in an article on my homepage