These are very common problems to have, even with large teams that include project managers. The first problem is prioritization. One way to deal with this is to do a weekly meeting to hash out the product roadmap and prioritization. If you do this well, the business will prioritize your work for you. To do this really well, you need a few things. First a rough sense of how much effort a project is. A day, a week? Second, the business should be estimating a rough impact on the business. A good question to ask is how much impact will this have on revenue? You should be working on things that have the biggest impact on revenue with the tiebreaker going to what's fastest to implement. Importantly you need to make it clear what your capacity is. My general rule of thumb is that you should be scheduling a maximum of 3 tasks to do each week.
Let's talk about the ill-defined tasks. The short-term solution is to say that every task requires at least one meeting to gather the requirements and design a solution. Until requirements are defined AND accepted by you, you can't schedule the work. You need acceptance criteria.
You're going to need buy-in from the business. Talk to your favorite manager/exec and discuss the problems you're having and your proposed solutions. They May have their own ideas and solutions to suggest. Trial it out and adjust the processes as necessary.
As a side note, you'll recognize some of these practices as agile practices. In factory terms, you're the bottleneck/constraint and you're trying to protect the constraint so that it's used as effectively as possible. Good luck
