Can I use the 2nd sentence instead of the first one:
I was swayed by her rhetoric into donating all my savings to the charity.
I was swayed by her rhetoric to donate all my saving to the charity
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Her speech failed to sway her colleagues into supporting the plan (Cambridge).
I think I can sway her to join our side (The Free Dictionary).
In light of these sentences, I think you can use either the construction sway + somebody + into + an -ing form" or the construction "sway + somebody + a to-nfinitive".
Furthermore, it's more appropriate and common to say savings instead of saving and charity instead of the charity in the sentence presented.
Both seem ok, but I prefer "to donate" over "into donating". If I recall both the infinitive form (to X) and the gerund form (X-ing) act as the same part of speech here, so it's more a question of personal taste and style than grammar.
It also varies depending on which verb or idiomatic expression is used. For example:
She convinced me to bake her a cake.
She talked me into baking her a cake.
"Swayed by her rhetoric" is an unusual enough phrase that there doesn't seem to be a preference one way or the other. But as written, it's an awkward sentence, which I would rewrite as a compound sentence:
I was swayed by her rhetoric, so much so that I donated all my savings to the charity.