Trying to Delay Joining without Losing the Offer Letter - Help!

Hey folks, need some quick advice! Got an offer from a company with a 6th June joining date. But here's the catch — my stocks (20% of my CTC) vest on 18th June. Missing them for just 10 days would really sting. How do I buy time without raising red flags? Should I say:

1. My current manager needs more transition time?

2. I have a family emergency? The HR’s already pushing hard for an early resignation and might suspect I’m still negotiating. Can’t mention the stocks (already used that angle for a hike). What’s the best way to handle this so I can delay joining till 20th June without spooking them?