I'm going to move this to IP for you so that the right people can answer this.
I am an Attorney-at-Law, licensed to practice law only in the state of California. Unless we have both signed a formal retainer agreement, you are not my client, and my discussion of issues does not constitute legal advice. Opinions expressed herein are those of the author, and do not necessarily represent the opinions of those who hold other opinions.
Wishful unrealistic "wants" don't get you very far. You need written agreements and the ability to enforce those agreements. A non-disclosure agreement (secrecy agreement), if mutual, might restrict YOU much more than the overseas manufacturer. Your information is largely public while the manufacturer's specific choices of production techniques likely are not publicly known. It is other forms of agreement you want. Joint venture agreements, exclusive marketing agreements, exclusive distribution agreements, non-circumvention agreements, exclusive manufacturing agreements, and exclusive representation agreements might be more valuable.
The use of a brand name will be a big long term protection, as your trademarks are possible to protect internationally and give you a commercial edge once your reputation is established. Until then you need other protection. You may have key improvement patents you could obtain as this business moves forward. I suggest you work closely with IP counsel to find or create those patentable improvements.
You need to work closely with a savvy business attorney and IP attorney or your fears are likely to become reality and your business to be stolen from you by both competitors and partners.
Joint venturing is a common way to contractually enforcement tie up a foreign partner. You appear not to have the buy fitness or legal experience to know how to properly joint venture to solve your situation. That's why you need a transactional IP lawyer.
I am not your lawyer and you are not my client. Free advice here is without recourse and any reliance thereupon is at your sole risk. This is done without compensation as a free public service. I am licensed in IL, MO, TX and I am a Reg. Pat. Atty. so advice in any other jurisdiction is strictly general advice and should be confirmed with an attorney licensed in that jurisdiction.
If you can get the proposed manufacturers to sign an NDA., it is worth the effort. However, most NDA's provide that there is no violation of the otherwise confidential information is available from public sources. Since the patent has issued, the disclosures there in constitute public sources, and anything in the patent would probably not be covered by the NDA. Nonetheless, experienced counsel know how to craft NDA's that deal with these problems and provide reasonable protection. Furthermore, when you develop your product in consultation with manufacturers, it is likely that improvements may develop which could be separately patentable. Even if some of the information about your invention is public, improvements and processes for making your invention may not be public. Thus, I encourage anyone in your situation to attempt to convince potential sources in the U.S. and abroad to enter into NDAs.
I am not your lawyer and this is not intended to be legal advice on which you rely. My answer is merely intended to assist you in understanding some of the issues that you face so that you can make an intelligent choice when you hire legal counsel.