We are performing experiments that use fluorescence resonance energy transfer (FRET) and fluorescence correlation spectroscopy (FCS) to monitor the movement of an individual donor-labeled sliding clamp protein molecule along acceptor-labeled DNA. In addition to the FRET signal sought from the sliding clamp-DNA complexes, the detection channel for FRET contains undesirable signal from free sliding clamp and free DNA. When multiple fluorescent species contribute to a correlation signal, it is difficult or impossible to distinguish between contributions from individual species. As a remedy, we introduce "purified FCS", which uses single molecule burst analysis to select a species of interest and extract the correlation signal for further analysis. We show that by expanding the correlation region around a burst, the correlated signal is retained and the functional forms of FCS fitting equations remain valid. We demonstrate the use of purified FCS in experiments with DNA sliding clamps. We also introduce "single-molecule FCS", which obtains diffusion time estimates for each burst using expanded correlation regions. By monitoring the detachment of weakly-bound 30-mer DNA oligomers from a single-stranded DNA plasmid, we show that single-molecule FCS can distinguish between bursts from species that differ by a factor of 5 in diffusion constant.