How do you deal with cold outreach reply overload without going crazy?

Thought getting people to reply to outreach emails was going to be the hardest part but site owners actually respond pretty fast.

The real bottleneck turned out to be managing hundreds of back-and-forth threads checking link placement and negotiating terms. Plus going direct ended up way cheaper than agency marketplace rates.

How do you guys manage high-volume email replies when a small team gets flooded?