It's probably the valve body or the clutch packs. It doesn't really mattter since either way you'll be buying or rebuilding a transmission.
-Redbeard
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Are u using manual or automatic transmission? Make sure if it's a manual, that you are pulling up on the spring for the reverse lockout. You didn't specify and not sure if you just bought the car. Took me a little while to figure it out too lol. -Drives3pedals
you need a new transmission
The most common problem for the 1995 Cougar xr7 Transmissions is the shift selenoids and the TCM Transmission control modules needing to be reprogramed common fixes for the shift control module would be a transmission overhaul or replacement do to the fact of the shift control module is located inside of the transmission and as well another cost effective fix is to get the transmission control module reprogrammed.
Shift solenoids are inside the transmission.
The gears in your transmission may be sticking or worn out. This would cause your transmission to not be able to shift.
I know you can get to 2 of them by taking off the front drivers wheel quite ez I think one is the shift select sensor
Shift solenoids are inside the transmission, and require disassembly of the transmission to replace.
There are several things that can cause your manual transmission not to shift out of first gear or reverse. The most common cause is a broken shifting fork.
the sunshell gear is stripped out
On a 1999 Mercury Cougar : Overdrive is the normally allowed position ( it won't shift into overdrive until the speed is faster anyway ) If you find that the transmission is frequently shifting between overdrive and drive ( such as hilly roads ) then it is best to use the transaxle ( transmission ) control switch on the side of your gear shifter to shut off the overdrive ( O/D OFF will appear in your gauge cluster )
if it is a aod transmission it is on the transmission usually a sensor near the shift linkage .if it is a Manuel it is on the top at the shift tower.
The neutral safety switch is located on the driver side of the transmission on the shelf part of the trans.. It screws down into the trans. and has a four wire plug connector.
You may have a broken spring which disengages the transmission allowing it to shift or it is bad linkage going to the transmission.