Would God use a child being born out of adultery as punishment for your sin and not forgive you?

No

A child being born as a consequence of adultery is rather a natural course of events. It might also be a punishment, but it’s probably pretty much a self-inflicted one in the first case.

As for never receiving forgiveness, the Bible nowhere says that adultery cannot be forgiven. In fact, no less than David committed not just adultery, but murder, and was still forgiven. But this didn’t mean that the logical and/or natural consequences went away, as David discovered to his pain. See 2 Sam 12:1-14 for the story.