My Dev sandbox seems to be API version 36 and our other sandboxes (and production) are API version 37.
This means I cannot deploy changesets into our Dev sandbox. Is there any workaround to do that
You cannot deploy components that are version 37.0 until your Production org is updated to new release .
There are certain components that you may downgrade (like apex,VF,components) using IDE and still deploy but for some components where salesforce does not provide ability to downgrade you will still not be able to move to PROD .
In case you are interested in Summer 16 release dates please see below
May 20th: NA44 and NA45 are upgraded
The first release weekend is when we here at Salesforce and any customers on NA44 and NA45 are upgraded.
June 3rd: The second release weekend
This is when another set of instances - NA6, NA7, NA8, NA9, NA17, NA18, NA22, NA28, NA29, NA31, EU5 - gets upgraded.