There is a difference between ownership in the LLC and acting as an employee of the company. As a member, you own an equity interest in the LLC, and unless there is a specific provision in the operating agreement, or a separate agreement, a member can compete with the LLC without it affecting them. Correspondingly, you can resign as an employee, you can't just resign as a member of an LLC. You can give up your LLC interest, or you can sell/transfer the interest (depending on the operating agreement provisions for transfer). If the resigning member wants a payout, and you deem it in your best interest to give him a buyout then that would be a negotiated issue since the operating agreement does not provide for it.
Regarding competing with the LLC: The provision that prevents this is called a restrictive covenant, as needs to be reasonable in a geographical sense (i.e. prevent competition only within a certain area) and for a limited time (i.e. for a 6 month period or 1 year -- you can't prevent competition for too long a period). The restrictive covenant should be drafted in order to assure that the NY courts will approve it (if it ever came to that). Usually that requires monitoring cases to see how courts are ruling. It usually depends on the industry, i.e. in the computer industry -- 6 months woud be a long restriction.
You may want to consider negotiating the payout with the resigning member and including a restrictive covenant (with non-disclosure provisions) as part of the buyout in order to protect yourself from competition.
DISCLAIMER—This answer is for informational purposes only and discusses general legal principles, trends, and considerations and is not intended as specific legal advice regarding your question. This answer does not establish an attorney client relationship. For legal advice, you should retain legal counsel in your state for advice regarding your specific circumstances.
In NYS, the default rules regarding the mechanics of an LLC are very slim (see the NYS LLC Law). Hence, most of the nuts and bolts of the member to member relationship are enumerated in the partnership agreement (i.e., the "operating agreement"). Hence, any restrictions against a member operating a competing business before or after membership is typically addressed in most operating agreements. And typically these clauses default to the following: 1) that during the membership term the member can "moonlight" in other businesses, but not competitive ones; 2) that post-membership there is no restriction on operating a competitive business. In the absence of any clause or other agreement that says otherwise, an ex-member typically can operate a competitive business. That being said, there are ambient rules about how this is done which involve fiduciary duty, trade secrets etc. that may or may not impact your situation.
Also, with regard to member resignation, again it is the operating agreement that dictates the mechanics (and often valuation) of how a resigning member's interest is treated. In many operating agreements, a resigning member forfeits his/her interest for gratis, while in others there is a discount penalty for resignation (the idea is to deter a member from "walking away" from the business).
You may want to have a business attorney look over the operating agreement (if any) to see what your options are.
I hope this helps.
Disclaimer: This answer is for informational purposes only and does not constitute general or specific legal advice, nor create an attorney client relationship.