Json-LD is indeed considered to be the "newest" protocol which is what i believe you are referring to as "new" and "old". This is a nice description of different types :
I'll cover Microdata and RDFa in same because they're similar. So they
both are addition to HTML attributes indicating what Schema.org field
names correspond with what user-visible text on the page. Works
perfect but need lot of developer work and designer changes. Because
both backend (admin interface) and frontend (HTML) must be changed.
And there are many issues that can be messed - incorrect
implementation, "rich snippet spam", software bugs, etc. Look easy as
1-2-3 but in reality it's pain (PIA) for implementation and support.
Example - only product implementation require in backend least 10 edit
boxes if they're manual filled.
JSON-LD - it's relative new protocol based on Schema.org. The main
benefit is that you split representation layer (HTML) from semantic
layer (JSON-LD). In prev. formats - they're same and linked each
other. Now they're split. This give you much more freedom than before.
You can place HTML data whatever you wish and just add hidden JSON in
head or in content that will add semantic marking. This is future (for
My personal preference is JSON-LD, reduced html bloat, ability to insert data without disrupting the flow of site is a must, its also the preferred protocol by Google which was confirmed by John Mueller back in 2016.
To answer your question though, if you wish to remove a certain part of the markup from your site, simply remove your values from Yoasts plugin and rely on your own JSON-LD alone.
For example the following screenshots should contain NO data if you do not want to implement Yoasts markup.
If you also have Yoasts breadcrumbs on your site, you can disable those ( if you do not need the user experience side of visual breadcrumbs ) and you could use a plugin like this :
To enable JSON-LD breadcrumb markup, without the visual bloat.