2003 Sequoia, possible failing oil pressure sending unit...

Hello, all:

I have a 2003 Sequoia SR5 4X4 (non-VVTi 2UZ-FE). I bought the truck in June 2018. She currently has 167,000 miles, but the original engine was replaced by the previous owner with a Toyota-remanufactured engine at about 113,000 miles. Apparently, the previous owner had taken it to Jiffy Lube and they forgot to refill it with oil. The engine was replaced at a Toyota dealer.

For the last two weeks since it's been getting colder (morning/overnight temps below 45 deg), I've noticed that my oil pressure gauge has been working intermittently. On a cold start, the needle will stay below the "L" and not move at all. After driving for about 10 minutes, once the engine is fully warmed up, the gauge will suddenly spring to life and start working normally. The needle then stays at about the 1/4 mark at idle, and smoothly sweeps up to the 3/4 mark when driving. Once it starts working, it doesn't bounce around or anything. If I shut the engine off and start it again within 10 or 15 minutes, the gauge will usually start working within a second or two of the engine starting, like normal. If it sits longer than that, it will take few minutes of driving to start working again.

As I said, I've only noticed this since the weather has gotten colder. I suspect that I have a failing sending unit that simply doesn't work until it gets warmed up, but I wanted to see if other have experienced this or have any insights. Could a dirty sensor be causing this? Can the sending unit be tested with a multi-meter? If it matters, I am running Quaker State full synthetic 5W-30, with a Toyota filter. I am due to change oil within the next few weeks.

Thanks!
Tim