An LLC offers a great deal of protection for the members against personal suit but the protection is nt absolute and the property owner has pretty much the same liability in their role as property owner that they would if you were doing this without forming a company. The best way to protect everyone is probably for the LLC to maintain a liability insurance policy that specifically names everyone you want protected as additional insured. I note that you say the classes are free of charge. It is not clear to me if there is any money involved in this business, be it through grants contributions or otherwise. As to what form of business entity would best serve your needs and what steps you need to take to best protect everyone involved, you really ought to discuss this with a lawyer and a CPA. You will not be able to get adequate information and advice through forums like this.
You can be sued personally no matter your choice of entity. Anyone can sue anyone else for anything. The salient question is not whether you can be sued, but whether you can be held liable on the facts that led someone to sue you personally.
Choosing a business entity warrants a very fact-intensive inquiry with your lawyer and your accountant, so make appointments with appropriate people. Liability insurance is advisable as well, so have a good long talk with your agent about your coverage needs.
Not legal advice, just my two cents. Consult NY counsel to obtain legal advice. I practice in Vermont ONLY.