I have stem, DT, and thumbie friction shifting bikes in addition to indexed DT, brifter, and trigger shifting bikes. I have to admit that shifting the friction DT bike with Suntour freewheel and power ratchet shifters is my least favorite. The ratchet action on the shifter doesn't quite seem to match exactly with a perfect shift. What I end up doing when downshifting is to overshift a bit until I hear the chain just start to engage the next gear then back off a bit. Upshifting is similar but I go up a gear then move the lever to downshift until I hear the chain then back it off a bit. Im not exactly sure if this is the same thing that others have described as trimming the gears, if it is sorry for my long winded explanation.
For shifting the front, it's basically just authoritatively moving the lever to move the chain and just trimming the derailleur to not rub. I find front shifting to be much easier. I actually prefer friction front shifting because the trimming adjustment allows for easier front derailleur installation. It just works and you don't have to fight with the chain rubbing on the derailleur in certain gears like on indexed front systems. As for which hand I use, I am lefty and use my right for both front and rear shifting with DT shifters.
I hate to say this, but what's with all the fear about riding with one hand? If you have a halfway straight frame, it should track pretty straight with no hands. You just have to lighten your grip and let the bike track. The light grasp on the bar is just in case you hit a bump or something so your bars don't twist. I have several bikes and they all handle different, but I practice riding all of them no handed at some point in every ride. Maybe taking up mountain biking could also help with your bike handling and confidence.