2 things contributed to his declining health. Firstly, when he was trying to kill Olivia, he was shot multiple times. This seemed to wound him. Secondly, as one of his shape-shifting colleagues says:
You've been in this body too long. You're dying. You need to get back home and get fitted with a new conversion device.
So, shape-shifting is a necessity for them, holding the same shape seems to have negative effects on them. Alternatively, the loss of the 'conversion device', which is the tool they use when changing shape, was the reason for the decline and not keeping the same shape. Perhaps the device helps maintain the shape-shifter in some way that wasn't clarified.
While drinking blood won't help you or I improve our health, putting the appropriate blood straight into our bloodstream would. However, the shape-shifters are complex biological and technological devices, not people, so it's possible they were manufactured with the ability to process mercury to heal.