If your ship has a target lock on it and either ship moves causing both ships to finish the movement phase outside range 3, do the target locks automatically come off as they are classed as out of range?

No, they remain until you acquire a new target lock/get destroyed etc

You need to be within Range 3 to acquire a target lock, but once you've got it, it can stretch from one corner of the play area to the other and you won't lose it.