I ran into a somewhat similar problem using HP ViziPrint Impress SEL in an HP Latex 330.
From what others have mentioned, I know it can be frustrating because the suggestions people are giving aren't what you are seeing.
But for me, it was an issue like others are describing. The printed media looked dry, looked great. But after spooling onto the take-up reel, when I went to unroll the drops to cut them (this project was a multi-panel window graphic), they were oily. I *know* that when I rolled the print up it was dry, but overnight it somehow gassed out or the ink wasn't properly cured.
Once I noticed this happening, I had to run the print again because the now oily ink smeared when touched. So this time I watched very carefully, and the print *was* oily coming out of the machine. I took a blow dryer to it while it was rolling onto the takeup reel. It again appeared dry, but after sitting overnight, it was oily again.
Eventually, I maxed out the inter-pass delay on the printer. I couldn't adjust the temperature because raising it would cause deformations in the media. However, increasing the number of passes and most importantly, maxing out the inter-pass delay did the trick.
Sure, it took a helluva lot longer to print those 7 drops, but in the end it worked.
The only other issue I could think of is your material isn't coated for Latex inks. But that doesn't seem to be the case because this issue is happening on multiple media types.
What is the relative humidity in your building? I know here in OK we had a monsoon last month, it literally rained every day. The humidity outside was 90% just about all month, and it no doubt impacted some of our material. It didn't help that the lady next door to us flooded and took her sweet time getting her carpets dried out..... :/
I hope something can help you. I know it's frustrating, but at least you have people trying to help.