You'll need to cancel your Plus card, and wait 30 days before opening up another Southwest personal card since you can't have more than one personal SW card at a time. Oddly enough, you can have more than one biz SW card, just not the personal ones. [ETA: well, it is possible if you do a MDD.] If you end up applying for the Plus again, just be sure that you haven't received the SUB on that card in the past 24 months. If you have, then apply for either the Premier or Priority.
It's based on anniversary date. Since Chase rolled out the new benefits yesterday and has sent out emails about them, I'm guessing they are immediately available for use.
If I'm understanding you right, you're planning on your DH getting the CP? If so, I think the best plan is for you to wait until the new SUB offer link is available from your SW Plus card (it usually takes a few days whenever Chase rolls out a new SUB on a product) and then have your DH apply through your link so that you get the 20K RR bonus. Use up your EBCIs for your return trip, because I agree you don't want to be worrying about that while you're on a ride.* When your card anniversary passes, wait for your anniversary points to post before canceling your card if canceling is what you want to do. You can cancel a Chase card within 30 days of the annual fee posting and get a full refund of the annual fee.
Now if you're thinking that you'll go for the CP yourself, instead of your DH getting it, then you for sure will need to cancel your card at least 30 days prior to applying for a new SW card of any flavor.
The advantage of your DH getting the CP is the extra 20K RR into your account if you can generate the support link with the 100K offer. I expect someone here will post when it goes live.
*You still need to check-in, but not exactly at T-24. EBCI boarding position will be assigned at T-24, and you'll find out what that is whenever you get around to checking in.