Multi-disciplinary teams need to be collaborative
There simply is no way around this. Teams cannot do their individual work inside a vacuum. Also, user stories should describe features, and should be developed in vertical slices, not horizontal:
e.g., your UI designer should not complete the UI design before discussing the designs with UX, web developers, and your DBA's. Multi-disciplinary teams need to be working together constantly to ensure the right product is being built (and the right product is being built correctly), furthermore, each team member should value and understand what feedback loops are and why they are important.
Your original approach to breaking out the work is correct
I believe you have the right approach and I have seen many successful teams do it this way. Although your approach is correct, I believe you need to re-focus your user stories on the features/functionality that is important to the user as opposed to referencing a degree of quality. I feel that quality is something the Product Owner and team should be addressing with Acceptance Tests as well as input that could be gathered from additional functions such as a user-feedback session.
Importance of Daily Scrums and Following the 3 Questions
It sounds like your team has trouble communicating and one discpline area is held up by another discipline area that causes the original work to become stagnant. If this is the case then the Scrum Master needs to be asking these questions on a daily basis, forcing this information out of the team, and working to facilitate discussions regarding these stories early and as often as possible in order to ensure that the team is properly collaborating and that blockers are being addressed.
Reiterate the importance of Feedback Loops
Scrum is a very simple process to follow, and at its core, it is simply an extrapolation of the Scientific Method. I think it's very important for every team member to understand this and to approach work with the idea that they need to apply this method, share their work, analyze their results, and update their approaches based on the feedback they receive with shorter cycle times being the most preferable.