Fortunately, I was taking a break when I read Tony's suggestion.
I took a little time and placed patches of sound deadener on broader sections on the underside of the dash. I used a heat gun and a stiff spatula to help adhere it to the surfaces.
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Some are hidden by pieces already assembled.
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I found hollow spots and such by shaking the assembly by the crash bar and knocking around the top of the dash. The noises began changing from a "tonk!" to a "thud". Time and trial will tell if it really works, but I think it will help a noticeable amount.
It occurred to me where the biggest echo chamber was on the dash and nipped it in the bud. The sound change after putting deadener here was drastic:
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(That is the underside of the instrument cluster hood.)