You're conflating liability with taxation. If you form an LLC, get the LLC a bank account, separate "finances", etc. and pay yourself out of that. They can "exist" simultaneously as your sole prop is someone paying you and 1099'ing, versus paying your LLC. You'll need a federal tax ID, and the proper paperwork with your state.
The difference is when you're sued, they're suing the LLC versus the Sole Proprietorship (aka you). If you separate assets then they can only get at the LLC.
If you "form an LLC" but keep using your personal account it would be tough to prove you've separated things.
On the tax front: If you have a single-member LLC and passthrough taxation it's all the same as someone paying you directly with a 1099.
If it's a different kind of LLC there are other tax situations, but it doesn't sound like that from your question.
Edit: because you asked, here are answers to your specific questions:
How is the transition most easily managed, from a legal and tax perspective? There is no "transition" process. File the state paperwork, make a federal EIN, and tell your customers to start writing checks to your LLC. Tax-wise, if you're single member and passthrough taxation, it's the same. It's just income.
Can both exist simultaneously? Yes. See my original answer. A sole proprietorship isn't a thing, it's just a fancy way of saying "someone paying you instead of a business entity"
How do I handle reporting both on this year's taxes? (Maybe two schedule Cs?) Again, just income.
How do I determine the salary or disbursements from the LLC? If this is a single member LLC then it's just reported as income on your taxes. It's not a salary or disbursement. If you're talking about paying employees and W2'ing yourself, paying FICA, etc... consult an actual paid professional.
Suppose I have no corporate expenses. Is there anything preventing disbursing the money from the LLC to my personal accounts quickly? Again no, see above. It's passthrough taxation. Any money the LLC makes is reported as your income. You're talking like you're going to keep money "in the business" and W2 yourself - which is a way it can be done - but then get ready for corporate income taxes on top of payroll taxes.