"In the packet Ways:" this mat be a double meaning- "packet ways" meaning the yards where packet ships were built, and or it could refer to a publication, "Way's Packet Directory", published by Capt Way (Frederick I think) that listed the history of all listed packet boats (boats that took freight as well as passengers), and was the yellow pages or bluebook for steamboats. https://www.amazon.com/Ways-Packet-Directory-1848-1994-Mid-Continent/dp/0821411063
If my second meaning is true, Bill is citing the Ways directory as the source of his info about Bonita's origin. "A Sweeney" in this context means an employee of or an actual member of the Sweeney family in Wheeling West VA, which for generations built steam engines and paddlewheel steam ships, most famously as "AJ Sweeney & Sons." It means that, as Bill tells it,according to Way's Directory, Bonita was built by the Sweeney shipyards in Wheeling and took two months to build (two months from her timbers till she moored).
"Paid the fare in billet"- means you paid no money for the trip,instead providing unpaid labor, usually loading / unloading freight. "A striker's boast would fain me most, about the wrecks the shoals were keeping"- Bill would love to be a striker (a term for an apprentice steamship pilot), thus being able to brag about how dangerous his job is and how pilots on a boat not as good as Bonita, end up wrecked on the shoals. "And how the old girl's got poor Billy's ransom saved." Following up on the last line- Bonita will keep him safe from the danger, no matter what. She wi;; bail him out, keep him out of trouble, pay his ransom.