Did Gollum deserve death?

[The Might]

As Mithalwen already pointed out, Gollum never was on an errand for Sauron intentionately.
This matter is presented well in the Hunt for the Ring chapter of the UT.
It is said that after Gollum was allowed to escape, he was followed by spies from Mordor, but that they were unable to follow him in the Dead Marshes. Afterwards he was captured by Aragorn, and taken north to Thranduil's realm.
If he had known that by seeking The Shire he would help Sauron find the Ring, he would have definitely taken much more precaution. Gollum definitely hated Sauron, and saw him as his greatest foe, if it was someone Gollum would have done anything to prevent from getting the Ring, it would have been Sauron.

One could debate that he might deserve death for other deeds such as killing Deagol, but definitely not for serving an evil power.