Q:

Resetting Soulbound

Say I have a creature with the Soulbound ability,  Stonewright, which is currently bound to my  Llanowar Elves. If I play a  Birds of Paradise, can I switch my  Stonewright to be bound to the Birds instead?

A:

NO.
Says right in the reminder text:

(You may pair this creature with another unpaired creature when either enters the battlefield. They remain paired for as long as you control both of them)

You may only pair when either enters the battlefield.
They remain paired for as long as you control both of them.
another unpaired creature implies the creature with soulbound must also be unpaired.

Of course, you can "reset" the soulbound by causing the card to cease to exist for an instant and re-enter the battlefield with, say, a cloudshift.
Another opportunity for pairing with another creature is of course if you lose control of the elves, if only temporarily; or if it leaves the battlefield.
These restrictions are made more explicit in the comprehensive rules where there is a lot more room to write out the two triggered abilities in full:

702.94a. Soulbond is a keyword that represents two triggered abilities. "Soulbond" means "When this creature enters the battlefield, if you control both this creature and another creature and both are unpaired, you may pair this creature with another unpaired creature you control for as long as both remain creatures on the battlefield under your control" and "Whenever another creature enters the battlefield under your control, if you control both that creature and this one and both are unpaired, you may pair that creature with this creature for as long as both remain creatures on the battlefield under your control."

