Well the first thing to realize is that it was a gradual process and secondly it was not that urgent because the dollar was a commodity currency and lastly bank notes were privately issued.

Let me begin with the middle point first, a dollar in the constitutional sense is 371 4/16 grains (0.77 Ounces) of fine silver, which was the measured average weight of the spanish milled dollar or piece of eight, which had been in circulation before the War of Independence and remained in circulation thereafter. Because there were these and other foreign coins in circulation, the dollar could serve as a unit of account even before the federal government started minting its own coins. The coinage act of 1792 authorized the following coins in gold $10, $5, $2.5, in silver $1, $0.5, $0.25, $0.1, $0.05, in copper $0.01, and $0.005. There were two ways these coin entered circulation. First, people could bring their gold and silver either foreign coin or bullion and have them minted into coins at cost. They would then spend them into circulation. Second congress would authorize the purchase of copper that was minted into cents and half cents (at a profit) and along with foreign coin received in taxes which would be re-minted into us coins these would be spent into circulation by the government.

Second, turning to the first point, this was a gradual process, foreign silver coins remained in circulation as legal tender until 1857. Congress gave these coins a value in dollars in accordance with their wight and fineness.

Third, banks charted by the states issued their own bank notes redeemable in coin. These allowed foreign coin and bullion to be converted into dollars without being re-minted thus more cheaply.