How long do I wait to refile again? After SSA changed my birthday. I called them and they said that they already updated to my corrected birthday.

Generally, it takes 3-4 weeks for the updated information to transfer over to the IRS.

But if you don't mind getting rejected, you can try as often as you like until your transmission goes through.

Do you think it's better if I send my paper return instead?

Generally, it takes 8-12 weeks to process a paper return and 3-4 weeks to process an SSA change.  Plus, paper returns are more prone to errors in processing.  Unless you owe the IRS and have to send a payment anyway, I'd wait.  You still have a chance to beat the deadline and if you have a refund, there's no penalty for filing late.

If you owe taxes, you can make a payment on the IRS website.