The blades vary massively in quality. I use astra, which are fancy German blades that run me like... 10 Aussie bucks for a hundred iirc (they're way, way cheaper online than in stores) and they're great. Feather I find even sharper but "grabby" and uncomfortable to use. My first buy was actually a sampler with a bunch of different brands but that was about a decade ago now, so I couldn't give you much advice.
Try both adjusting the razor and different blades. That said, most people use cartridge razors just fine. I feel like safety razors, straight razors, and shavettes (my choice these days) are mostly about ritual, although if you're lazy about shaving like me straight razors and shavettes don't clog up BUT you run a higher risk of cutting yourself. Personally although I do a good job sharpening knives I wouldn't get a straight razor as they sound like a bugger to maintain and are expensive for a good one.
I love that her plan is on 1" squares as per a DnD combat map.