New FamilySearch and FamilySearch are two different entities.  new.familysearch.org is currently only for LDS members.  If you have a FamilySearch.org login, it will not work with New FamilySearch.

FamilySearch.org is the database of historical records that have been transcribed or filmed and posted by FamilySearch. This site is open to anyone. By contractual agreements some of the microfilmed documents are not available for viewing by some or all. There is a way for people to view some documents that others cannot and that is by getting involved with FamilySearch indexing. (indexing.familysearch.org). You build up points by volunteering your time to transcribe the microfilmed documents. Some of the contracts allow people who participate to view their records. There are also some small family trees available here. Creating a login account here will allow the system to know that you are also doing indexing.

So using RootsMagic to do WEBSEARCHes using FamilySearch Records is available to all. But currently if you are not LDS you can turn off the new FamilySearch features (LDS) within the TOOLS > File Options. When new FamilySearch goes live to everyone then you can turn the appropriate tools back on.