The process of modifying text has gotten harder in Rhino 6 due to the use of rich text (RTF) and the fact that you can change fonts and font styles in a text string.
If text has a font set by an override for part of the text and you want to change the text content, you have to make RTF that includes the font override.
If the override is for the whole text, not a subsection, you could get the override style used by the current text and use ON_Annotation::SetOverrideDimensionStyle() to apply it to the new text.
You can tell if the font is overridden by using bool ON_DimStyle::IsFieldOverride(ON_DimStyle::field::font font)
And const ON_DimStyle& CRhinoAnnotation::GetEffectiveDimensionStyle(const CRhinoDoc* doc) to get the style, including overrides, from the current text.