I have selected text wrapping around my image. I only want text from 1 paragraph to wrap around the image and the next paragraph to start below the image. How do I do this without adding blank lines to "push" the new paragraph downwards?

IMO, the best method would be to put your image and the related text into a two-column table without borders.
Insert the image as in-line-with text inside the first table cell.

Word will let you wrap text around an object but you don't really get much control over the text flow. Therefore, the workaround suggested by Charles is the way to go.

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Stefan Blom
Microsoft Word MVP

Microsoft 365 apps for business
Windows 11 Professional