There is a chance the Sox never made a formal $300M offer, but it was floated in negotiations as a starting point (or might have been the end point.) The response or nonresponse by the Betts negotiators might have given the Sox a clear impression, their offer was not even close, and they decided to move on right then and there, but like I said, who knows.
Your final statement seems very plausible, but I will add, trading Mookie might not have really been what the Sox wanted. They likely wanted him to stay for $300M/10, but once they knew that was a no go, they "wanted" to get what they could for him, and dump Price, at the same time.