I'm not entirely sure that I see the problem with ending up in crucifix after a shoulder mount... unless you were just trying to go into a straddle shoulder mount and hold it there first?
It sounds like the biggest issue here might be that you are relying on momentum to get into both of these. Once you have that momentum going, it can be really hard to stop it (so you keep going past the intended point).
Building your core strength will absolutely help with these because you need core strength to be lifting up into these moves instead of using momentum to get you there.
I can see that you subscribe to Veena's lessons, so I'd suggest looking through them to work on your core strength. And I'm sure there's a shoulder mount video that might help explain your problem too. It's been a long time since I've been a lesson subscriber, so I can't refer you to any specific ones.
I struggle a lot with consistency in my shoulder mounts. I totally understand the urge to use momentum to get up there. Try to think about really tightening your entire core and tucking your pelvis forward to get those legs up, instead of swinging a leg behind you and then kicking up.
For the one-handed handstand, both core strength and balance training should help you.
Sometimes I find that working the reverse of an exercise is really helpful to me. For example, you could invert normally, go into a crucifix, tuck your upper body to place the pole in the shoulder mount position on your trapezius, then slowly release yourself into a straddle shoulder mount hold and work on a should DISmount from that position. Slowly and with control.