、。。
God, you're persistent, Shinobu. I think I'm about to cave.
Literal
Really persistent [feeling-なぁ], Shinobu-[polite-さん]. Kinda worn-down-by-persistence [seems-そう].
根負け (literally 'defeat of the root') is an idiomatic noun-vs-noun for being worn down by someone else's persistence until you give in — a compact word for a specific social dynamic English handles with a phrase. しそう attaches そう 'seems / about to' to the masu-stem of する, forming 'seems like (I'm) about to X.' なんか 'somehow / kinda' is a casual hedge softening the following assertion. Note the dropped particles throughout — characteristic of relaxed conversation.