To clear the connections in the rest API you will need to do a "DELETE" on the site-session:
DELETE /qcbin/rest/site-session
Cookie: LWSSO_COOKIE_KEY={cookie}; Path=/
Cookie: SiteSession={cookie]; Path=/
You can find this in the REST API reference guide under "How to do common tasks" then further under the "session management" heading:
As for why it was just causing a problem now it is hard to say, my recommendation would be to add the site parameter REST_SESSION_MAX_IDLE_TIME as mentioned in the above page, the default is 60 minutes but if you are having issues with idle sessions making ALM slow you may want to lower the number to like 15 or 20 minutes.....