There are a couple of issues involved in this.
The first issue is whether or not your neighbor's surveyor and survey are correct. The only way to challenge or confirm the survey is to get a survey of your own (It takes an expert to challenge.). If you do not want to incur this expense, then you will probably need to concede that your neighbor's surveyor and survey are correct.
The second issue is whether or not you must move your fence. Whatever she told you is probably not important in this context. What is important is that, if her survey is correct or you choose not to challenge it, you will need to move the fence. You are trespassing on her property. You have not had a fence on her property long enough to adversely possess her land -- and, from the facts as you have related them, you did not intend to adversely possess her land.
If your fence was built in reliance on the work of another surveyor or other professional, you may be able to recover the costs of doing so from that person. You will probably need a lawyer to recover, however.
Good luck to you!
The material on the site is made available with the understanding that this author is not engaged in providing professional legal advice. Before relying on material on the site users should obtain appropriate professional legal advice. The material on this site may include opinions, recommendations or other content from third parties that do not necessarily reflect this author's views. Links to other Web sites are included for the user's convenience and do not constitute this author's endorsement of the material on those sites, or any associated product or service. The listing of a person or company in any part of this site in no way implies any form of endorsement by this author of products or services provided by that person or company. The questioner is advised to consult an attorney for legal advice as to that person's particular situation. You may wish to consult your lawyer or another lawyer instead of me (us). You may obtain information about other lawyers by consulting the Yellow Pages or by calling the South Carolina Bar Lawyer Referral Service at 799-7100 in Columbia or toll-free at 1-800-868-2284. If you have already engaged a lawyer in connection with the legal matter referred to in this communication, you should direct any questions you have to that lawyer. The exact nature of your legal situation will depend on many facts not known to me (us) at this time. You should understand that the advice and information in this communication is general and that your own situation may vary.
My colleague is correct and you plan of action is reasonable. Builders can make mistakes, but your builder likely had a plat of your lot prior to building the house and the fence builder would have been on notice.
If you are in the wrong and can keep the fence in place you may wish to purchase that tiny strip of land from her. That purchase may have to be approved by the appropriate zoning official or property owners association, but could save you money on moving the fence, and keep your title clear of the encroachment on her property.
This answer is for general advice and does not create an attorney client relationship with James Mosteller or the Mosteller Law Firm LLC.