1992 Toyota Celica Oil Pressure Problem

1992 Toyota Celica GT Convertible
2.2 5SFE 201,000 miles on car and 80,000 miles on used engine
5 speed manual transmission

I have a problem with my Celica which I just bought. I just got a tag on it this past Monday and have been able to drive it. It does have a bad oil leak but not severe. I haven't had a chance to fix it as I had to leave it at the trim shop to get the top reconditioned. I suspect it's coming from the oil pump. The engine was replaced with one from a local junk yard with 80,000 miles on it. I do not know how long this engine sat before being installed in my car. I drove the car to my moms house probably 10 miles away and on the trip back (about 19 miles into the round trip) the oil light started to flicker. It seemed to flicker more as I accelerated. I went home and checked the oil. It was to the full mark. The oil was just changed and has less than 30 miles since changed. I went ahead and changed the oil filter and the oil pressure sending unit since it was cheap. I drove the car about 5 miles and it started flashing again but much sooner than the first time it did it. I'm using 10w 30 oil. I can turn the car off and wait just a minute or two or just 30 seconds and start it back and the light is not coming on but after a couple minutes of driving it will return and slowly begin to flicker until it stays solid with random flickering. I can't use my oil pressure guage as none of the fittings fit this engine?