CPS Problems?

Alright, I've searched and I think I've figured out my problem, just needing some expert opinions:
1997 Country 4.0 with automatic, 104,000 mi.
I've only had the vehicle for about 1 month. When we test drove the vehicle initially it stumbled at highway speed a few times. Took it back to the lot and told them about it and they sent it to their "mechanic". Supposedly they found no problems, which I kinda believe because it hadn't done it again since we bought it (until a few days ago). Sometimes stumbles on acceleration, sometimes stumbles at highway speed, sometimes has trouble cranking. Initially, I thought it might just be where the car lot had pressure washed the engine, causing an intermittent electrical problem until it dried out. Now, reading here (invaluable resource by the way), I thing it is probably the Crank Position Sensor. Am I thinking right or is there something else to check?

CPS normally causes a no run problem. More likely is that when they pressure washed the engine they got the TPS throttle position sensor wet. This is a fairly common problem.

CPS was the weak spot on older MFI engines but not so on the newer ones.

With your system you need to check the code from your computer first.

I don't think it would still be a moisture issue, it's been about a month and it's hot as hell here

Engine codes (from on-off method)
12
11
55 (end of test)

11 No crank reference signal detected during engine cranking. Intermittent loss of either camshaft or crankshaft position sensor. CKP sensor target windows have too much variation.
12 Direct battery input to PCM was disconnected within the last 50 Key-on cycles.

CPS???

Question:
I think I've figured out that it is indeed the CPS that is my problem, but why do the symptoms seem to be worse when the engine is cold? At first, all I noticed was an occasional stumble. Now, it goes stone cold dead (to the point of having to be re-started) sometimes several times before it gets warmed up. Why would a sensor such as this be sensitve to the engine temp?

The cps can cause it to run bad. When I got my 87 It would Idle great, and go up to 15mph fine. them it would start to stumble and shake. No one could find the problem, thats why they sold it to me for $700. new sensor problem was gone.

My 90 had some run/starting issues when I acquired it from my youngest son (180k). Things kept slowly deteriorating and starting issues increased until one day it wouldn't start. Changed the CPS, all the problems went away. While not simple to change, it sure isn't one of the harder things to do on an XJ, nor is it one of the expensive things either.

If you have doubts about it being the CPS, I'd just change the thing. It isn't going to hurt things, and if it doesn't solve the problem, at least you know you have a new CPS in it.