If you want to fund your own position, probably the best course of action would be to look for the various postdoctoral fellowships awarded by organisations like the NSF, the European Union, and many other national science funding bodies. Your advisor may know some examples that would be most appropriate for you at this career stage. Each fellowship of course has their own rules, so you would need to study them carefully. Generally, the fellowships at this stage would support yourself through either a stipend or by providing your institution with money for your salary. They may or may not pay overheads. In addition, there may be a modest research budget, which for theoretical fields typically can be used to cover for example conference travels.
In case of the fellowships that I am personally aware of, you would have to mention a host instution at which you would want to carry out your research. Typically you would have to provide a signed letter of commitment from the institution, agreeing to hire/support you if your fellowship would be awarded. So indeed it is important to get in touch with possible institutions. You would usually also have to argue in your application why this place would be a good fit for the project.
So how to go about once you have found suitable funding sources? As you said, it is indeed a good idea to contact professors at those universities that would be a good fit for the project that you have in mind. Your advisor might be able to help here as well in suggesting and/or contacting the right people. At this stage, a full research proposal is not really necessary yet, but you should be able to explain briefly what you are interested in, and how it fits in with the group.
Once you have found someone to support you, you can work on your proposal according to the guidelines for the fellowship. Your prospected host instution may provide some feedback on the proposal, but in my experience this is more common only for bigger grants at later career stages. In my experience, professors are generally happy to support your application, since it is relatively low risk for them. But be sure to secure their support in time, since it may take some time to navigate the bureaucracy to get a letter of commitment, for example.
You can only apply for each fellowship once per round, but you can apply with a similar proposal to different fellowships. In some applications you will be required to disclose this, however. And almost certainly, if you would get awarded more than one fellowship, you would have to choose between them since you cannot get funded for the same research twice.
Since typically the institution has to provide a letter of commitment, once you are awarded a fellowship they will be obliged to hire you (or support you, in terms of office space and such) for the duration of the project. At this point it will be mainly up to the adminstration of the university to proceed and prepare the relevant paperwork.
Good luck!
