Upon hearing the tragic news, she almost burst into tears.

Literal

Tragic-news [object-を] heard [then-て] she [topic-は] tears [object-を] spill looked-like became.

涙をこぼしそうになった means 'almost spilled tears' — ~そうになる means 'almost did, was on the verge of.' 涙をこぼす is a literary expression for crying.