This is really the best answer here. Really good advice.One of the reasons that DE/SE and straight shavers get a better shave lies in their shave prep.
Your “splash face with hot water a few times” technique is not much of a pre-shave routine, but as you use shaving brush and shaving soap/cream already, giving your face a little more time to absorb some water, maybe with some (glycerine) soap, to soften the whiskers may help. The pre-shave does not have to be elaborate, but it is essential that the facial skin has a few moments to soak up some water.
Another reason for DE/SE/straight shavers having better shaves lies in the way we shave.
To get the close and comfortable results that most of us are after you would have to abandon one-pass shaves.
A one-pass shave and BBS shave are just not compatible.
You would need to raise your ante and do at least a two-pass shave, plus touch-up on the corners of your yawline.
With the right safety razor (positive blade exposure) you could even mimic the results close to a straight shave.
But then again, I believe that with a good cartridge razor, you could mimic the results of a good safety razor shave, if you are willing to abandon your one-pass routine, even if some on this forum may disagree with me.
The catch is that cartridge razors try to emulate a traditional multi-pass shave by having multiple blades, but you are still dragging that thing in a single pass and in the same direction across your face. There is a risk that you could give yourself skin irritation with a multiple pass cartridge razor shave. That is the reason why DE/SE/straight shavers harp about “comfortable” shaves.
There is obviously a reason that you abandoned safety razor shaves in the first place. If it was the number of passes and you don’t want to change, then a saftey razor will not give you any better shaves.
To summarize, what will give you better results is a) an improved pre-shave routine and b) shave with multiple passes. You could first try with a good cartridge razor and then, if you are so inclined, move from there to a safety razor, but I am confident that you will already see improvements with the cartridge razor.