Hard to say what swallow down means without more context. It could mean she ingested it, took it into her.
"Right up to" means something goes to that limit. Not that it was her only her right side. Her arms go the whole way to her shoulders. Her feeling of the cold did not stop at her wrists or at her elbows.
It went the entire way up her arms, or "right up to" the shoulders.
"We walked right up to the edge of the cliff."
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