While Paddle Steamers still operate on the Mississippi river, these are tourist only; gone are the hey-days of hundreds of them churning up the river silt.
Historically, many of them blew up when their boilers exploded, others were wrecked in skirmishes, crashes, or being caught in battle. Possibly the most famous (and least noticed, as it happened just after Lincoln's assassination) was the SS Sultana, killing 1800-2400 people when one of the boilers exploded on the Mississippi river. The wreck still exists:
In 1982, a local archaeological expedition uncovered what was believed
to be the wreckage of Sultana. Blackened wooden deck planks and
timbers were found about 32 feet (9.8 m) under a soybean field on the
Arkansas side, about 4 miles (6.4 km) from Memphis. The Mississippi
River has changed course several times since the disaster. The main
channel now flows about 2 miles (3.2 km) east of its 1865 position.
It's noted during the 1880s, at least, many wrecks lay on the river floor. However, as the previous quote illustrates, the course of the river has changed with time, so some are no longer on the river bed and presumably have been easily cleared away. Others would also be removed as they pose hazards to other vessels using the waterways, and others would have rusted away over time.
For example, one of the more famous ones was La Belle, or La Salle Shipwreck:
La Salle's French expedition of 1684-1687 tried to locate the mouth of
the Mississippi River, but went too far, ended up in Texas where this
15 m long 6-gun frigate sank in shallow water in 1686. The lower part
of the hull was found in 1995, excavated 1996-97, by draining the
wreck area with a cofferdam. Conservation still continues.
As a result. it's dubious whether any wrecks remain, and if they are, they're almost certainly not "explorable", unfortunately.
While not a paddle steamer, one of the famous civil war boats was the ironclad USS Cairo, the wreck of which was discovered in 1956, and is now on display in the Vicksburg National Military Park.