I can't help but notice a lot of similarities between your team and my own during our first year, and others I've had the chance to interact with. You have a group of passionate students, but they don't all see your vision. They love FIRST and want to do well, but don't seem to understand what it takes to move to the next level.
Not having an "adult in charge" is very difficult. You seem like you have a strong idea of what you're doing and are probably far ahead of many student leaders of teams. However, without an adult there to drive accountability and handle a lot of the background logistics so that you can focus on leading your team, your attention will be divided.
Your teacher's "hands-off approach" was probably well suited to FTC, but the stakes are different now. Decide if participating in both FTC and FRC is appropriate for the team at this stage - you may be spreading yourselves too thin. While students change year to year, she is the constant - if she is as committed to the success of robotics at your school as you are, she will need to modify her role a bit. Sit down with her (perhaps with some parents too) and try to identify how she can best help the team. This may mean taking a more active role in the team structure, keeping students accountable for what they need to do, and serving as a "safety net" before things get too out of hand. It takes time to set up a solid team structure, and I applaud you for your effort; see if she can help take it the rest of the way. The issue of your subteams "blurring together" and lacking definition is not unique to your team and something I've seen before. Help each leader be accountable for their area of responsibility by outlining clear job descriptions and using a system like Trello (or a simple whiteboard) to track the tasks their committee must complete.
Don't feel guilty about the grants. You earned them - grants aren't handed out. They were earned by your team. The fact that an adult helped write the applications for a young team is not a bad thing. It will take some time before your fundraising committee can be independent and entirely student run. Be thankful for the grants, keep in touch with your sponsors, let them know how much you appreciate their support. This will set a foundation for successful, long-term relationships with them.
Over the next 29 days, work on preparing your team for success. Organize the workshop, appoint a head of "operations" to keep inventory and keep everything organized. Purchase whatever parts and tools you know you will need. Outline clear job descriptions for each subteam leader. If possible, create a handbook that outlines team rules and procedures that all team members must agree to (
see our example and many others here on CD).
Don't give up. You have very high expectations of your peers, as you should. The rest of the students are certainly capable of what you ask of them, but they have to want the same thing as you. Try to communicate this message with the help of your teacher. Make the most of this season, and try to share your story with as many students at the school as possible so that you have a continuing group of students. This won't be your team's last year, it will be the first on the path to greatness.