Not sure if still relevant for the OP, but as I lack modesty I'd like to say that it is possible, to a good approximation, to hedge varswaps dynamically using 3 delta-hedged options only. This is something I found out relatively recently, and explained in the following note:

https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4542475

The static portfolio of a continuum of options (and a dynamic position in the underlying) for varswaps replication is beautiful theoretically, but not feasible in practice.

Volswaps is a different matter, the square root really messes up things, and I am not sure there can be an 'easy' hedge for the volswap.

Not sure if still relevant for the OP, but as I lack modesty I'd like to say that it is possible, to a good approximation, to hedge varswaps dynamically using 3 delta-hedged options only. This is something I found out relatively recently, and explained in the following note:

https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4542475

The static portfolio of a continuum of options (and a dynamic position in the underlying) for varswaps replication is beautiful theoretically, but not feasible in practice.

Volswaps is a different matter, the square root really messes things up, and I am not sure there can be an 'easy' hedge for the volswap. As the volswap can be regarded as derivative on the terminal value of a varswap, it can be written as a static strip of options on realized variance (a la Carr-Madan), but not as a static strip of options on the underlying asset.