There is an account of Thomas and Royal Hazelton traveling to the goldfields of California in a Greene County, Missouri history book.
My curiosity was sparked to learn more about the events that took place for the Hazelton men during this time period. I searched on Ancestry.com and found one passenger list with two Hazelton men aboard, R H age 18 and A age 35. They were traveling from Chagres to New Orleans in March 1851 aboard the Brig Roselin.
This page doesn't state the date, the previous image in the set is dated March 7, 1851. The National Archives website states that record group M259 is a chronological set of records. So we assume the date is close to March 7, 1851. The Royal H Hazelton who died at Baton Rouge would have been 65. His sons, Royal H and Alpheus are most likely the men on this passenger list.
Chagres was the river they traveled on to cross the Isthmus of Panama. Once they reached the Pacific Ocean they would have traveled North to San Francisco. A google search returned this result: http://www.maritimeheritage.org/ports/centralAmericaPanama.html with transcriptions of news paper articles that are quite revealing!
I hope to find more records in Louisiana, Panama, and California to reveal more about my Hazelton ancestors during the California Gold Rush.