I'd say sitting & drinking are gerunds (used with the preposition "instead of"), the 1st one is the -ing form of the progressive form.
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AnonymousHe needs to be fighting instead of sitting in a Russian palace, drinking martinis with President Putin.sitting - gerund
LinguaphileDrinking is also gerund hereAnonymousHe needs to be fighting instead of sitting in a Russian palace, drinking martinis with President Putin.sitting - gerund
drinking - present participle as a qualifier
AnonymousHe needs to be fighting instead of sitting in a Russian palace, drinking martinis with President Putin.fighting: present participle