I figured out a work-around when I was copying text from another Google Doc that already had h1, h2, etc that didn't fit into the target document.
If you highlight text, in the right-click menu, there is "Select all matching text". This selects all text with the same formatting in the document. For instance, if you select an H2 text, it will select all H2 texts. If you select bolded, italicized text, it will select all bolded, italicized text.
First, what I needed to do was cut the copied text and paste it into a new document. That's because I didn't want to select H1s in the whole document; I just want the H1 in the copied text. So I needed the copied text alone in another document.
Then, in that new document, I selected an H1, right-clicked, and selected "Select all matching text". This highlighted all the H1s in the copied text. I changed them to H3 with
ctrl+alt+3, which is what I needed them to be in the new document. Same deal with H2s, demoting them to H4s.
Then I cut and pasted the demoted text back into the target document. Done!