I don’t expect the number of blades is critical compared to whether the blade(s) be, well, razor-sharp.
I find a good shave is mainly a matter of beard preparation. Straight blade razors are okay but I often end up with a small cut. Electric razors just don’t shave close enough. I don’t notice a ton of difference between 3 and 5 blades.
What works best is filling the sink basin with warm water, as hot as comfortable. Take a flannel, towel or facecloth and wet it, and hold it against your neck, chin and cheeks for a couple minutes. Take the shaving foam and spray a small pile into the warm water. After removing the towel, apply warm shaving foam to the face. You could use gel or soap, but I don’t prefer it. I like using bristle brushes but they don’t actually add much to this method.
After shaving, brush out the hairs with a toothbrush; or wash them out. Let the blade dry by putting it upside down in its case.
Doesn’t scrubbing, rubbing, or otherwise touching the blades trash the keen edge? They are not made of diamond or carbon nitride, though I am willing to believe, considering the cost and the hard usage people are reporting, that there is some kind of decent alloy involved.
A double-edged safety razor will give you a much better shave than any cartridge razor. And replacing the blades is dirt cheap.
But before you buy a new razor, as others have said above, work on your prep.
Use lot of water. Make sure your razor is wet and clean with each stroke.
Over time learn your beard. Touch your face to figure out which way the hair grows on each sector.
Eventually three passes should be ideal—with the grain, across the grain, and against the grain. You might be lucky to need just the first two.
Take a hot shower, giving your face plenty of moisture, and wash your face with a moisturizing soap.
After showering, don’t dry your face completely. With your face still a bit damp apply a pre shave lotion or oil as soon as you get out of the shower, and then you can do other stuff while that lotion soaks into your face.
Use a shaving cream/lotion/soap that lathers well and really massage it into your face. Don’t just pat it on and leave it at that. A shaving brush is great for this, but some people don’t want to get into that yet.
Yes, if I’m in a hurry, I get a washcloth and soak it with the hottest water I could tolerate and press it against my face for even just 30 seconds (repeating it as it cools off). Just a bit of normal soap across my face (doesn’t matter what kind – bar soap, liquid soap, it all works for me), and I’m good to go. For me, it definitely seems to be the heat that does it. I don’t need more than 30-60 seconds of this type of prep and my whiskers are soft and easy to shave. Without this, it is painful as all hell.
I can and do use pretty much anything I find a good deal on so I personally don’t have a good recommendation, however this article from Men’s Health singles out the Schick Xtreme 3 and Bic Twin Select disposables.