I've been compiling some Burke County Land Grant abstracts that we published in the Burke County Genealogical Society's journal over the years. (I'm the current editor.) The name Lambeth's Fork appears a couple of times in the timeframe you mention. In one of them, the "Lower Little River" is mentioned. This generally identifies the area of eastern Caldwell and Alexander Counties today. There are three 'Little Rivers'...Lower, Middle and Upper. Powell's North Carolina Gazetteer (UNC Press, 1968) gives more information on their locations.
The parties to the grants appear to be the same, Robert Holmes and Patrick Sloan. They are given as neighboring landholders and even chain carriers for the surveys. The Burke County Public Library in Morganton has copies of all the Burke Land Grants (taken from the originals in the State Archives in Raleigh). The two grant numbers are 454 and 506.
Here are the abstracts (as published in the Journals):
454. Patrick Sloan 300 acres on both sides Lambeths Fork adj. Robert Holms. Entered 28 May 1778 #302. Surveyed 28 Aug 1779. CC Robert Holms and Adam Hall. Grant #456 28 Oct 1782
506. Robert Holmes 365 acres both sides Lambeth's Fork Lower Little River adj. Patrick Sloan, George Brown and Simon Walker. Entered 10 Sept 1778 #69. Surveyed 24 Aug 1779. CB: Robert Sloan and Adam Hall. Grant #508, 28 Oct 1782.