WAAS and DGPS are different technology with the same application. DGPS beacons were mostly coastal or "Special use" and were a tower that broadcast a position correction based on a surveyed position. This waws than transmitted and GPS recievers with a special DGPS antenna would collect both signals and apply the correction.
A lot of units were DGPS capable but all required the extra antenna. No handlheld unit that I have ever seen actually had the antenna. The DGPS antenna on my boats was quite large.
When the USA switche off SA the DGPS beacons were also switched off.
WAAS uses a pair of satelites over the continental USA to broadcast a time correction signal. The unit then uses the corrected time to amend the almanac that it downloads on startup and thus correct the standard GPS signal.
A DGPS signal is more accuracte if you were going to supplement the signal for a precision approach as it can also fix height data as well due to a known position.