This will get gimmicky and old pretty fast if you're not careful, and you run the risk of burnout.
That said, there are some ways to make it work.
Pre-plan a few lines you know will get used.
Prepare some fairly generic rhyming words you can use in a pinch to put a few lines together.
Do not solely use couplets! Mix up your rhyme scheme with limericks and other more-than-two-line rhymes.
Finally, and most importantly: When a rhyme would be super-forced, you can't come up with anything, and/or you're a few sessions in and it's starting to get old: Describe the rhymes instead of saying them. This is a trick I suggest often to roleplayers, who tend to get stuck in a "RPing means I have to say everything my character says" mindset, which is entirely not true.
Instead of trying to find a fifth rhyming word for the name of the town you're traveling to, "[Character name] mentions that the stone bridge should be the right way. [Bard and/or high-Linguistics character name] doesn't think the word she rhymed with Sandpoint is an exact rhyme, but it's probably not worth starting an argument over." works great.
[Witch] uses a subtle double entrendre to remind the party that she's pretty much out of spells. [Chaotic character] has to repress a smirk.
The crone's face indicates intense concentration. Either her true seeing spell has uncovered something very important, or she's still trying to find a word that rhymes with 'Urgathoa'.
And, of course, when she absolutely must break character:
[Witch] sighs with resignation. "Why the [expletive] [deity] does it have to be orange?"