Ahh, I've seen this problem before. It seems the vast majority of people at my gym need help to get the first rep up on dumbbells and I know why. You partially figured out the problem and started doing your reps close to how I prefer to do them.
This is how you should db press to avoid those 2 problems you're having. Lay back with dumbbells, rest them on your chest, tuck your elbows to about 45 degrees to your body and slightly rotate your hands so the dumbbells are also at about 30-45ish degrees to your body. When you press up, you can rotate your hands back the other way so that the ends of the dumbbells are parallel to each other, squeeze your chest at the top, then lower back to the starting position.
I've had a lot of guys at my gym start dumbbell pressing in this manner compared to the elbows flared, not coming down all the way that you mostly see (were probably doing yourself) and they ALL have said it feels significantly better on their shoulders, they feel it in their chest more and have been coming back to me saying their chests are growing more.
Give it a try, it should help you build the strength you need to always get the first rep on your own, alleviate your shoulder issue, as well as result in better chest development. If you need a spotter on dumbbell presses, you're doing something wrong, IMO, of course with the exception of doing shoulder presses with 100+ plus dumbbells where it's just a pain in the ass getting them up to the starting position.
On a side note I'd highly recommend you do the DC stretches to stay flexible/stretch fascia as well as the broomstick/towel stretch to decrease the likelihood of shoulder issues.