It appears that you tend to be a perfectionist. Aren't you? Well, that's one of the challenge you got to face if you are a perfectionist. In adverse situations, the team members might also play games. That's natural in a highly political environment.
Is being perfectionist your problem? No. Then, how will you overcome the situation. There are ways and means to control the situation and it's purely in your hands. It's only you. Does it sound like a self-help cliché? Yes, it is.
Does it hold true for all those perfectionists? Quite naturally. Let us analyze the situation further.
Consider you assign a task to your team member. You would assume that by assigning a task to your teammate, the individual would take the ownership and revert to you when complete. If this is the case, then, your expectation is very much correct. That's how your team members would need to function. But you need to understand that they are not "YOU". They are different and have their own characteristics. You will need to accept this and change your mindset. This is the first step to success. Do you agree?
Okay, I have changed / ready to change my mindset. Now, What?
Well, there are umpteen ways to attack this problem. Instead of enlisting a huge set of possible solutions, let us narrow down to only two areas. What would it be? It would be: a) hard-way, b) soft-way. These two "ways" shall be your highway to your success. Crap, is what you will think. No, it is not. There is no other option other than to keep reading. Please bear with me.
Okay, so what is that hard-way? It is all about preparing the hard deliverables. What else, it would be your "task sheet" also known as Work-breakdown-structure (WBS) that spells out who does what and when. But then, you would ask, I have the task sheet prepared. Now, what?
The answer to this question would be: If you have prepared this WBS all alone thinking that your team members will need to go by your schedule, than I am sorry. You are at fault. You are preparing to blow your own trumpet. Trust me, that happens. I have committed that mistake and have burned my fingers and toes.
You should possibly involve your team members, brainstorm (yup, that's the buzz word), and then arrive at the WBS. The next activity shall be to communicate. But to whom? To the stakeholders. Communication is key. Please note. The moment you broadcast, your team will feel the heat. They will get to know that the tasks are getting noticed from top to bottom. Use the communication tool wisely.
The next set of activity shall be to keep monitoring and tracking. This is very essential. Publish the status religiously - rain or shine. This means you will need to over-communicate regularly until you gain traction.
Figure out the results. You will be surprised to believe the story. You can create history.
Now the soft part. This is very essential. You know that "give respect to take respect" is a famous adage. But then, that's what everyone expect. Please look at your team members as equal. Never indulge into any gossip. At times, just by using the magical words (please, thank you, etc), you could sense where your project is moving to. Always, be calm. Never vent. Your behavior at work depends on your project and your professional success.
Soft skills are very important to delegate and to get the work done. You might sometimes need to keep prodding. That happens, but then, as a PM, you got to see yourself as a guide.
Hope this helps you.