How do I remove disgusting Emojis?

If by "PO" you mean "previous owner" (and not "program officer" or "parole officer" or "post office"), yes, that's possible. It would also mean that the phone was not properly prepared for you, the next owner. This could lead to other problems.

You should make sure that these things have been done:

What to do before you sell, give away, or trade in your iPhone or iPad - Apple Support

You may need the help of the previous owner for some of them.

fv74 wrote:

I did the transfer and kept most of the original Apps.

It would have been far preferable to erase the phone completely before making it "yours".

I have tried the (...) icon numerous times but it won't let me select the offending emojis. I can change them temporarily but they come back the next time I use the phone.

You should not "select the offending emojis", quite the opposite in fact. Start selecting different emojis, those you find acceptable and in time the offending ones will be hidden.

There should be noting left for the Previous owner on a device if it was correctly prepared for sale or transfer.

With that said, the poop Memoji is technically built in, and you can change it by tapping on the (...) icon there and selecting a different Memoji type from the list then tapping on a specific Memoji from there to use. As you use other Memoji's, it will replace the ones there with others you use. You don't need to send them, just tap a Memoji so it's added to a message and then tap on the (x) on the corner to remove it from the message. It will replace one of the Memojis with the one you selected.

If the previous owner did not erase the iPhone entirely, you should make sure your Apple ID is signed in to iCloud on the device, and turn off find My, then go to Settings ➜ General ➜ Transfer or Reset iPhone.

You should never use a device that was not properly erased by the previous owner.

Try a Forced Restart, it resolves many issues.

Force restart iPhone - Apple Support

If that does not help, about your only option is to erase the phone and start as new. You can redownload the apps. Keeping the old ones and not erasing your phone was not a good idea. I hope you are using your own Apple ID.

fv74 wrote:

I am curious why not erasing the old Apps is a bad idea? I have not had a single issue with the exception of the turds.

If these old apps were ones the original owner downloaded under their Apple ID, 1) you don't legally have the right to use them under your Apple ID and 2) you won't be able to update them.

If they are the pre-installed apps that are part of iOS, then there is no problem.

One other thing you might try is resetting all settings. Personally, I think restoring the phone to factory state is easier but if you're not ready to do that, try this. Settings>General>Transfer or Reset iPhone>Reset>Reset All Settings. I think that should set the emoji back to their original state. It will also reset any other customized settings.

Your choices are simple, erase your phone and start as new (what you should have done originally) or live with the phone as-is.