The interpretation of lyrics and poetry is not simple nor is it cut-and-dried.
There is little or no punctuation, but I think we can consider 'I've drowned and dreamt this moment' a complete sentence within one verse.
Clearly we're dealing with metaphoric language, since a person can't claim to have their heart burst and then sing about it.
I've drowned means something like "I am completely overcome." I've dreamt this moment means that the speaker has had a dream and pictured this time when she felt completely overcome.
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