I get much closer shaves with my Merkur 33c safety razor than I did with cartridges.

I'll suggest another order: go 1. with, 2. across, then 3. against the hair. Against is the most skin irritating and largest risk of nicks, so may well be done with the least amount of work.

Also, warming the area to be shaved with warm water (or shower as you mentioned) does a lot for the closeness of the shave, don't skip out.

I find brush and lather mostly to make shaving smoother and lessen skin irritation, not make for closer shaves, but your mileage may vary. Also consider warming your blade (I run it under the warm water) between passes, I find it also helps for the smoothness although can't say why.

If you have to apply force to the cutting (rather than letting "gravity" do the work) you probably need new blades. They are supposedly single use, but I can typically get 2-3 shaves in on a regular blade, and even up to six shaves with some blades (Korean Dorco platinum is my favorite).

Adding force is the only way I get nicked nowadays, but it goes fast when I do, so I make sure to switch blades even between passes if need be.

Also, depending on your face and skin, you might want to try a safety razor with a different blade angle, there are handles that give a (slightly) more aggressive shave — meaning closer shave but (slightly) higher risk for nicks and irritation.

Best of luck!