Whether you blame Stackoverflow or not, your team has learned that there are no negative consequences for missing deadlines nor are there benefits to meeting deadlines. They get whatever they have coming no matter what. This is the first problem you have to solve.
I suggest setting some shorter deadlines and providing consequences for them not being made. The key here is to set them before client or major project deadlines are missed. Waiting too long will just make you look bad.
You have some of the responsibility but your team needs to learn that they can complete quality work on time or suffer the consequences. Unfortunately, you can't look over their shoulders and force them to work. They created the lack of trust, but you let them.