(apologies for earlier incomplete reply)
I think you did not interpret RFC 5763 correctly.
To quote RFC 5763 section 6.7.1:
When ICE is being used, the ICE connectivity checks are performed before the DTLS handshake begins. Note that if aggressive nomination mode is used, multiple candidate pairs may be marked valid before ICE finally converges on a single candidate pair. Implementations MUST treat all ICE candidate pairs associated with a single component as part of the same DTLS association. Thus, there will be only one DTLS handshake even if there are multiple valid candidate pairs.
The text does not say that all connectivity checks must be completed. In fact it expressly notes that the IP address may change after the first DTLS packets have been sent. At least one connectivity check must complete, but the DTLS exchange can start as soon as a valid candidate pair exists.