Hi Mary-The information that you have regarding your ancestry seems quite plausible.The DORANS were a very well known Traveller family from the Wexford/Wicklow area of South-East Ireland.They were renowned for their musical ability .They were Pipers,known for their expertise as UILLEAN PIPERS(Irish Bagpipes).Felix Doran & John Dorans recordings may be accessed online.
Fury/Furey is a very common surname in parts of County Galway.There are are still some Traveller families of Fureys in East Galway.More than likely your ancestors met at one the large fairs where the Dorans would have been playing music on the street-Modern day "Buskers".
Verifying your information will be very difficult.Compulsory registration of Births/Marriages&Deaths only began in 1864.
Baptismal records kept by parishes began around 1800.These are available online at NLI/Parish Registers but are sometimes incomplete and you need to know the parish or diocese.
Travelers/Tinkers were itinerant Dealer& Tradesmen who traveled around the country dealing in Horses and repairing /manufacturing Domestic and Agricultural utensils for Farmers & Householders.The mainly used Tin for Buckets,Jugs etcetera.Hence the name "Tinker" although today Traveller is most often used.Tinkers is considered somewhat derogatory.They became Travellers when many of them were evicted from their holdings durng the time of the Great Famine or earlier.
Contact East Galway Family History Society ,Woodford,County Galway-who may be able to help you