NOTE: Lots of innuendo, some swearing.

Before diving forward on this topic, I want to caution that any study of obscenity in language has a very hard time stating when the "earliest" a thing was, because of self-censorship and regular censorship. We know people have used swear words for a very long time but they were considered inappropriate for children and feminine ears and so were kept out of print and generally only used in "masculine" spaces; for example, with the Hundred Years War where the British regularly called their foes goddems. In a discussion of the word Uranus I noted in an earlier answer a joke (amongst gentlemen only) recorded in a 1859 diary as

If you put your head between your legs, what planet do you see? Uranus.

but the same meeting, when mentioned in print, went with a different joke entirely.

This means that it is possible the first obscene tongue twister might have been devised quite shortly after their invention in the early 19th century, but we don't have record of it until later. In fact, the earliest I have been able to localize having an obscene tongue twister in print is 1952, due to a combination of form of the tongue twister and the obscene novelty song. Let's start with the twisters first:

Andrew Airpump ask'd his Aunt her ailment;

Did Andrew Airpump ask his Aunt her ailment?

If Andrew Airpump ask'd his Aunt her ailment,

Where was the Ailment of Andrew Airpump's Aunt?

That's from the book Peter Piper's Practical Principles of Plain and Perfect Pronunciation, circa 1813, and yes, that's the Peter Piper, the one that still survives immortal to this day:

Peter Piper picked a peck of pickled Peppers:

Did Peter Piper pick a peck of pickled Peppers?

If Peter Piper picked a peck of pickled Peppers,

Where's the peck of pickled Peppers Peter Piper picked?

The book (by Willard Johnson, Philadelphia) is intended to, as the preface indicates lead to a "Profitable Path to Proper, Plain and Precise Pronunciation." Mr. Johnson immediately disappears from history and seems to have done nothing else of note; essentially anonymized, with Peter the surviving vestige.

Tongue twisters -- not yet known by that name, which only appears in the 20th century -- appear after in many pronunciation guides and manuals intended for children. This is the era where "she sells sea shells" also appears for the first time. In some cases books are very systematic and pedagogical, i.e. "exercise on collision of consonants the same in kind, but different in power", which includes

Red tape, old tea, bad toffy, good tin, sad tease, field tent, buttered toast.

Tongue twisters essentially remained a pedagogical exercise; as I already cautioned, there is little doubt some clever 19th century child discovered a loophole that is lost to time, but where we really have record would have to wait until later.

Switching over to obscene songs, we really do have examples quite early, even some in recorded format; the recent record Actionable Offences: Indecent Phonograph Recordings from the 1890s is dedicated to obscene songs and poems, as collected by Bruce Young.

My wife and I just stared at each other in disbelief, we were just amazed that that kind of language — what you think of as very naughty late-20th-century schoolyard talk — would exist in the 1800s.

Tracks include "Learning a City Gal How to Milk" and "Michael Casey Exhibiting His Panorama" and no, it isn't just innuendo, and yes, you can find excerpts if you Google them.

As I hinted at earlier, soldiers at wartime were a common source; I've written about obscenity in WW1 before, including this song:

Oh Kaiser Bill is feeling ill, the Crown prince he's gone barmy,

And we don't give a fuck for old von Kluck and all his fucking army.

Although possibly closer to our goal topic is something like "Hitler Has Only One Ball" which was popular amongst WW2 soldiers (you can listen to a modern rendition here). Other WW2 songs include "There's a Shortage of Good Whores in Mobile", "I Put My Finger in the Woodpecker's Hole", and rather directly, "For They were Large Balls".

So now, we get back to the obscene tongue twister song, and more specifically, the University of Michigan. There was a folklore project which collected songs, and multiple students registered "Sarah", a song that was apparently intended to be sung drunk ("popular in the off-campus tavern crowd"). One student noted:

If a person can sing this without a slip, he is not yet deemed inebriated by companions.

I'll reproduce the entire song (and the tune it goes to) in a follow-up comment, but the important line to grab is

Sarah, Sarah, sitting in a laundry shop,

All day long she sits and slits ...

which you can either test out inebriated or just try to do many times fast to get the idea. Yes. A harder variation (in one of the recorded "alternate verses") includes "she sits in her slip and sips Schlitz". This particular tongue twister became common enough it gets discussed in multiple folklore journals, and, the "sits and slits" even shows up in a volume intended by the Peace Corps to use with children.

But what about ships? That doesn't seem to have been discovered until later; we have Dr. Seuss in 1979 teasing us (in Oh Say Can You Say?)

If you like to eat potato chips

and chew pork chops on clipper ships,

I suggest that you chew

a few chips and a chop

at Skipper Zipp's Clipper Ship Chip Chop Shop.

and I haven't seen a reference until the 1996 Barenaked Ladies album Born on a Pirate Ship which, if you thought might be coincidence, makes things rather certain with the cover. By the lag time of this kind of lore it has to date at least the 1980s, and possibly the 1970s. However, "My father works in a ship yard" does appear roughly around this time, indicating that the pirate variation that everyone seems to known may have come from a derivation of that. All the different tongue twisters drop and and words as the teller desires, adding and removing deviousness (and a dose of Schlitz) as much as desired.