As Jon notes, there should be no "convergence" to just use the remaining link because both links are in the route table for the same destinations. I.e. once the router removes the one link, all traffic will shift to the other.
The forgoing (e.g. shut interface) should be the situation on a router where you shut an interface.
What could happen, though, is OSPF could continue to route to the dropped (failed) link during the time OSPF doesn't know it's down. If OSPF is signaled with a hardware link drop, it should be around 50ms or less. If the link loss detection relies on a timer, you could have traffic sent to the down link for multiple seconds.
You mention doing this "gracefully". If done correctly, your transition time will depend on how fast you configure both neighbor routers, but you shouldn't lose any traffic.
(How to do gracefully? Change OSPF cost on one router's link. Confirm that link dropped from ECMP. Shut link interface on other router. Confirm all okay. [Suggestion, only shut interface on router "nearer" you. You might also reset cost on first router now, so you need not remember to do it to later restore ECMP.])
