Learn Turkish: Tartışmak yerine, yeni bir öneri sunmanız daha faydalı. - Instead of arguing, it is more useful for you to present a new suggestion.

This is an action subject clause! If you want to say "presenting is useful" in general, you would use sunmak. But here, you are saying "your presenting is useful". To tie the action to a specific person, we use the short verbal noun -ma and add the possessive suffix -nız (your).

If we use sunmanız for the second action, why doesn't "instead of arguing" use the same ending (tartışmanız yerine)?

The phrase -mAk yerine (instead of doing something) is a standard pattern that almost always uses the full infinitive -mAk. Because the sentence already makes it clear who we are talking to with sunmanız, Turkish doesn't need to repeat the "your" on the first verb. Using tartışmak is more natural and avoids repetition.

This sentence doesn't seem to end with a verb. How does it work?

This is a noun sentence. The entire phrase Tartışmak yerine, yeni bir öneri sunmanız acts as the giant subject of the sentence. The predicate is just the adjective phrase daha faydalı (more useful). There is an implied "is" at the end, so it literally translates to: "[Your presenting a new suggestion instead of arguing] is [more useful]."

Even though sunmanız has noun endings, it still has an object (yeni bir öneri). How can a noun have an object?