cooling help needed 95 windstar

Hopefully somebody can guide me here Im at a total loss.
First I have a 95 Windstar with a 3.8.

The other day it overheated got towed back to house. Found that the thermostat was not operating. This led to a major problem due to snapping of one of the bolts then in the attempt of getting the bolt extracted cracking the housing itself. So replaced the intake and all gaskets and the thermostat. Started van fans were not engaging, Replaced coolant switch. Idled in yard for 30 min didnt overheat. Went for a test drive and within 3 miles of the house temp went from M to above the H in a split second. Had a friend tow me home with a tow chain.
With this I figured I might have a blown head gasket. So I checked the oil and there is no signs of coolant in the oil. There is no oil in the coolant.
So I then thought maybe there was a leak in the exhust side and when under highway speed was causing pressure to build back into the system causing the coolant to come out the overflow jug. Put a pressure tester on the system, Found no apperant leaks while it was going to 16 psi. I then let it sit under pressure for 2 hours and the pressure did not drop. So to me this tells me that there is no head gasket leak.

So as a last resort changed the water pump and both hoses just to know in my mind that this could also be eliminated.

Now it will sit in the drive and idle for an hour and temp will not go above the M the entire time fans are engaging, but if you go out on the road and accelerate it takes about 1/2 mile to a mile to overheat.

Please help broke student needing his transportation back.

I have been thinking this one over.....don't have any good solution.
What comes to mind is........Are you loosing cooant?.......also....maybe the system is drawing in air......air in the system WILL cause the problems that you are seeing.

Check to see if the upper radiator hose is full of coolant by squeezing it.....
Sometimes....if air gets into the sytem......the overflow tank will be full.....and air will be in the hose.....even a low coolant level in the radiator while the overflow is full.

IF you are getting air into the system.....check all the hoses for tightness (don't overtighten)......it would be strange for the system to let air in.....but not coolant out......but worth a check.
Air could come in through a head gasket leak.......which a test for combustion byproducts in the coolant would show.
The coolant in oil and combustion byproducts in coolant are tests that would require you to send in a sample of each to a place like blackstone labs to have them run the tests.

I would go for the testing before throwing a head gasket at it........due to the $$$ and labor...and chance to create more trouble in the process.......it pays to know that it is really the problem first.

After fixing all of that, are you really sure it is actually still overheating or maybe you are now getting a false reading from the temp sender to the dash gauge? With the new fan sender switch, are the cooling fans coming on as they should? Have you tried pulling the connector to the fan sender switch with the engine on while cold and the fans come on? They should. If you have an ohmmeter, you can measure the resistance across the temperature sender for the dash to see if it may not be accurate. It is a single wire unit beyond the thermostat towards cylinder three screwed into the intake manifold. The resistance may be going to zero at too low of an elevated temp indicating at the dash gauge a false high temp. I think the sensor or sending unit is under ten dollars if you would just rather replace it than worry about testing it. Be sure to buy one with a brass base rather than plastic, which have a history of not lasting long. To test it, I'm kind of guessing on this as I don't have access right now to my CD manual, but going by the values for the two wire engine coolant temp sensor, I think you can reasonably be able to check it. At cold engine temp, I'd expect around 40,000 ohms (40k ohms). It will gradually decrease as the temp increases. You can use as a rule of thumb for the engine temp is when the radiator fans come on at low, the temp when they go to high speed, and I think the thermostat will open just after the fans first come on. So from 180 to 220 deg F, you should see 3800 to 1800 ohms. If it is much lower than 1800, that would most likely be your problem. Just as a simple test for comparison, if you take off the single wire and ground it to metal on the manifold, etc., the dash gauge should go all the way to the hottest gauge position.