I think I can tell you what is causing this, not having oil changes regulary and using good oil. I've seen this lots of times, and performed a cheap fix dozens of times that has worked 90% of the time.
Here's what happens. Cheap oils break down and become varnish and sludge much faster than quality oil. This sludge and varnish will gum up your oil pathways and pressure sensor. Mustangs don't usually suffer from this too much because everyone knows they are performance toys, and so the oil used is better and changed more often.
I'll use my friend to show something: He has a 96 Dodge truck, same problems as you. He told me about it last year so I took a look at it. When I took the oil cap off the sludge build up was totally unbelievable. I put my finger in there and pulled a huge old goober of gunk out. Worst I've ever seen. Usually it's just varnish that can get you.
Anyway, we got 9 qts of cheap oil, 5qtz of Amzoil, 4 new oil filters and two cans of engine flush. First we put on a new filter and ran a can of flush in there, then changed oil and filter and ran the other can of flush. Both times we drained it was like, just disgusting... Anyway, put in the other 5qts cheap oil and filter again. Ran this just to purge any left over chemcials. Drained and put in other filter and Amzoil. Result: A much happier driver and much better running truck with no problems!
A lot of people will advise against using engine flush. A lot of people are gung ho for it. All I can tell you is that I have used it on engines for over 11 years (I flush mine once every two years just for preventative maint.), including all my own vehicles and lots of friends vehicles. I've heard the warnings of oil leaks and all sorts of stuff people like to say, maybe I'm lucky, I don't know, but after 11 years using this has done nothing but good by me and my friends.
If anybody else has experience with engine flush, good or bad, this is the thread to post it.