A client is paying monthly - I signed an agreement earlier in the year under my name and I am receiving the money to my personal bank account under my name. If I get an LLC (taxed as an S Corp), will I need a new agreement or anything like that? Or is it enough to change the direct deposit to my LLC's bank account? Or is it not even necessary to have a bank account for the LLC?
(I am not a lawyer.) In the US, an LLC is a separate legal entity from you. You signed the agreement, not the LLC. An LLC would do well to have its own bank account so that funds are not co-mingled. Co-mingling funds is one of the ways that the corporate veil is pierced, and you lose the protection of an LLC.
You might do well to chat with a business lawyer and determine your need for an LLC. If you really need it, then set it up with legal and accounting advice. In certain circumstances, spending the money on good lawyers and good accountants can save you a lot of money.