If a bra fits well most of the support comes not from the shoulder straps but from the band, especially if there are underwires. Shoulder straps act mainly to stop it from slipping downwards with gravity. The bra you bought had the rubberised bits to try to stop that. Look for another style/brand that doesn't have the rubbery grips but go down a band size so that it grips tighter around your ribs, and up a cup size to compensate. The actual cup will be the same.
What size are you? Bigger girls generally need more support.