I assume you plan out pages before creating the final pages, like smaller thumbnails before starting on the final thing. If you don't, then I would start there. Making thumbnails of pages and placing text bubbles in those stages makes thing easier down the line.
Another solution would be changing the shape of the bubbles and not having them be perfect ellipses or circles. If you placing speech bubbles digitally, then finding a way to draw them digitally may help in giving you the opportunity to make room for the words.
Another thing you can do in type the words before putting in the actual speech bubble. That way, you can already see how much space the words will take up before putting in the actual bubble. Now you're not cramming words into a bubble, but making a bubble big enough for the words. Also it's good to have the font size, depending on the font but for most standard speech bubble fonts, to be in the 14-18 point size. 12 can be a bit small, but you could possibly get away with it.
Lastly, you need to be alright with your art being hidden. Unless it's important to the scene, then just let it be covered by the words; it'll be fine.
And after looking through the comic you're referencing, your comic not, Jenny Clements, there were a lot of places where you could've given more room for the text.
Oh, one more thing. While I understand it be a stylistic choice fore some people, don't be afraid to have speech bubbles break the border of a panel. In your comic you do allow this, though very tepidly. Allowing bubbles to overlap with other panels will obviously give you more space. Though you have to be careful so not to confuse the reading order of the text bubbles.