There is no perfect solution unless you use a machine as they have much closer tolerances. However a few ideas to consider.
1. You should only make 1 pass on a cut, always, if you don't, you will see the layers in the bevel of the cut. No way around it. Make sure you cut goes through into the under layer board and penetrates it but does not cut it enough to bend. If it does you blade depth is too much.
2. You should replace the blades often, blades are cheap, mat is not especially if you are using rag or rag alpha mix.
3. Check the machine for true, my Fletcher 2200 needs to be trued about every 3 months or so, I am also an occasional cutter of 4 ply as I now use only 8 ply and I can't cut that.
4. Your corner needs to over cut on the back around 1/8 of inch.
5. Consider stops, I use them all the time on my cutter as once they are dialed in, things get much easier
6. The Logan 450 I owned was never true with the extension, may have been damaged during shipping. So straight cuts were never that good. I have wall cutter for that, but I realize you don't want to go there.
7. On straight cuts, check the cutter to make sure the blade holder is solid. The 450, it just runs on a track as I recall, it's not running on tube rail, but I may have it wrong.
8. For the sizes you are talking about consider using a ruler, get one of the rulers that has the vertical bar in the middle that you can both push against and protect your hand. This are much better for longer cuts IMO.
9. On the vertical cut, make sure the blade is new, as nothing dulls a blade faster than straight cuts.
The 460 is a longer version of the 450, and I have used it before. It should get you there with a bit of work.
Paul