I have two textboxes linked together. However, I don't want to break up paragraphs. So if a paragraph overflows from textbox 1, I want all of it to end up in textbox 2.
Yes you can actually. See Adobe's instructions: https://helpx.adobe.com/indesign/using/formatting-paragraphs.html
Use Keep Options in the Paragraph panel menu (if you have text selected, then click on the paragraph panel expanded options for this menu to be accessible). You can also do this within the paragraph styles.
Check "Keep Lines Together" and then "All lines in Paragraph." Start paragraph should be "Anywhere."