The Wilderland climate is probably continental, yes, with very cold winters and very hot summers. Although
the Brown Lands look like a very dry place,
Mirkwood must be moister. This wood is in a plain between the
Anduin (where there's some
marshes, like the Gladden
Fields), the Forest River and the Celduin, so there's no shortage of water.
Meantime, in the real world

D), you can see vast forests in Russia on
the east of the Ural, in Siberia, which is an extreme continental climate.
On a smaller scale, in France, there are forests east of the
Massif Central, where there is no oceanic climate, but a mild continental climate.
However, geology and climatology weren't
Tolkien's strong points, we all know that... Which is a pity, really, but then, perfection doesn't belong to this world !