In Canada, if someone from BC (12% sales tax) purchased goods in AB (5% sales tax), it's pretty much just that - you saved on taxes. Same as a Canadian going to the US to buy goods - either the border guard asks you to pay the taxes and/or duties, or they wave you on.
However, in the US, the use tax laws put the onus on paying sales tax on the consumer. So even if you were not taxed at point of purchase, there would be an expectation to, as noted, self declare and pay applicable sales tax, as if the purchase had been made in-state.
The only "out" I see for the OP is if their state, like WA, considered coins to be sales tax exempt. In which case s/he could simply say that they should had been classified as coins rather than antiques, and that as such they are exempt from state sales tax. If their state, like NY, considers coins to be sale-taxable, then they're pretty much out of luck and should pay the amount due.
(again, this is not legal advice)