I’m not an immigration lawyer, but as a lawyer here’s what I suggest.
First call the lawyer’s office. I would explain to whoever answers the phone that pretty much what you started your post with (“I’m a foreigner currently on exchange visa that expires in a little less than a year, and the situation is complex enough that I need some professional advice”), that Mr/Ms Lawyer was recommended by Ms. Whomever, and you’d like to speak to Mr/Ms Lawyer to see if he/she could help. Trust me, lawyers love to get calls about potential new business, and will usually be happy to respond.
Depending, the lawyer may want to have a brief phone conversation and/or have you in for a consultation. Many (though not all) lawyers don’t charge for an initial consultation, which is usually more of a meeting for the lawyer to understand the issue, see if he/she can help, and convince the client to hire him/her. At the initial consultation, you should discuss fees and billing. In many states (including New York, where I practice) lawyers are required to give new clients a written statement describing what they are being retained for and how they will charge. I will ask my clients to sign a copy of my retainer letter agreeing to its terms.
As for how much it will cost, I have no idea, but you should discuss it with the lawyer in advance. Often, with a complex problem, the lawyer won’t know in advance, but should be able to give you a good idea.
The obligations of a lawyer to a client is a highly complex area of legal ethics. However, the most significant obligations are that a lawyer must represent a client zealously (within the bounds of the law), in the client’s interest and without conflicts of interest, and that the lawyer is bound to keep the client’s communications seeking legal services confidential (with limited exceptions in some states such as to prevent a crime from being committed in the future).