I'll be honest, once it's in the embossing (texturing) of the product it's pretty difficult to get it removed. As you've already discovered, doing so by hand is nearly impossible. The reason is, in order to get the residual up and out of the texturing you're going to have to use a microfiber mop head rather than a traditional cotton head mop, a sponge mop, or a scrub brush. The fibers or bristles on those are too large to get down into the texturing. You have to use something smaller than the texturing to get the remaining residual up and out of the embossing. String mops, terry cloth, sponges, and scrub brushes are simply going to ride over the high spots in the embossing and you MAY get a little of the residual up, but it won't get all of it.
If it were me, I would here a company that does professional cleaning and would have access to a small auto scrubber. If you insist on doing it yourself, you can rent one, but just a heads up that there is a bit of a learning curve to using one.
The absolute best product to use as far as a chemical to remove the residual would be ammonia and water. Anything else is going to have other chemicals that can add another layer of complexity to the chemistry experiment you've already got started. This will not hurt the floor finish at all. Any remaining residual ammonia will simply evaporate when drying. You can buy ammonia at Dollar General for a couple of bucks for a whole gallon.
The cleaning solution should be 1/2 cup of ammonia to 1 gallon of clean water.
The advantage to the auto scrubber is that it can apply the solution, agitate it with the microfiber, and then after dwelling for 5-10 minutes remove the solution along with the residual chemicals using its extracting mechanism.
Since you've added other chemistries into the mix, it may take more than one shot to get it all up.
An even more IMPORTANT next step ---If there is any carpet in your house IT MUST BE CLEANED AND EXTRACTED AS WELL, or you might as well not even begin. Murphy's will track off of a hard surface and stick to carpet fibers. Then when you walk back onto your hard surface you'll track it back onto the floor. EVERYTHING MUST BE DONE AT THE SAME TIME or you're just going to track the residual right back into clean area and have to start all over again.
Murphy's oil soap, as the name suggests, is a soap. So, chemically it's a surfactant. Surfactants work by clinging to and trapping dirt in their "sticky" chemistry.
Once you get everything to a reset state, moving forward you can use a mixture of 1/4 cup of ammonia per gallon of hot water. As an alternative, you can use Mannington's Ultraclean spray cleaner. It's a good cleaner, but it's pretty expensive in comparison to ammonia and water and it won't yield any better result than ammonia and water.
A note on steam mops...
Yes, I know, I know, I know, Mannington says their product is steam mop safe. So do most other manufacturers. So, it must be safe right??
Let me ask you a question and see if you think using a steam mop makes sense to use on a plastic floor....
Steam is generated at 212°F...Mannington states very clearly in their installation instructions "Adura Max should NOT be exposed to temperatures greater than 140° F."
While they do state in their warranty that Mannington approves the use of steam-based cleaners specifically made for hard surface flooring", there are several exclusions in the warranty that reference "burns" or "heat" related damage.
Question...
What do you think possibly may happen to plastic flooring that isn't supposed to be exposed to temps above 140°F when you expose them to temps at 212°F???
Ultimately, it's your decision to decide what is best for you. I'm just trying to provide as much science-based information for you to make the best decision possible.
At any rate, what I've stated hopefully helps you resolve your issue. This information is culled from dealing with Mannington products both commercially and residentially for the past 30 years and performing these steps myself to help customers who have made the same error. It's not going to be fun or easy, but I know from experience this will work to correct the issue.
All the best!
CFR