You can define a new paragraph style in the Engrave menu > Paragraph styles (amazingly enough!)
When you hit shift-X to enter staff text, set the paragraph style in the box at the top left of the formatting options, then type the text.
You can’t really “define a new text style,” because Dorico tries to understand the semantics of most of the text in the score, not just the visual appearance. Things like “dynamics” etc might or might not be displayed as text (e.g. the same dynamic marking could be displayed as “cresc” - with or without a line after it - or as a hairpin). The only “non-semantic text” options are shift-X for staff text and shift-alt-X for system text.
This is fundamentally different from Sibelius or Finale. In Sibelius a dynamic mark might be a piece of “expression text” like ff (and you have to press the right keys to get it formatted correctly) or it might be a “line” like a hairpin. In Dorico, dynamics are dynamics.
You can change the font style for dynamic text, in Dorico but you can’t change a dynamic mark into something that doesn’t use the “dynamic text” style - and conversely it wouldn’t make much sense to try to use the “dynamic text” style for text that wasn’t meant to be a dynamic instruction.
Presumably, once the scripting language gets fully operational, you would be able to define a keyboard short cut that does “Start staff text and set the paragraph style to whatever” with one key press - but you can’t do that yet!