The more...the more structure with normal clause?

grammaticality

Solution 1:

It seems like there's another implied the more... before the second clause

The more they sing, [the more] their burden lightens and their love deepens.

This answer explains the syntax of these expressions.

However, I've never heard of omitting one of the mores in such an expression. If you don't want to use that format, you could say:

As they sing, their burden lightens and their love deepens.

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