I'm devastated -- my lovely 'Gift' A-1 is now letting me down when I use it with ambient light -- all exposures are about 5 times OVER-Exposed -- I did a Kodak T-max 400 and all were black on processing in ID11 1+1 -- I have de-sqeaked it about a year ago -- do you think some of the oil had got into an exposure circuit board -- no sign of oil when I took off the base-plate though ?
Today I did another test with 1991 dated Ilford FP4 film -- I set 100 ASA as it was old then did a series of 'Auto' exposures : 'P','AV,'TV, and I got 1/125 @ f6.7 on 'P' and 1/1000 @ f2.5 on the others which agreed with a Sekonic Spotmeter reading for 1/1000 -- so I took another on 'Manual' exposure with the lens OFF the Green 'A' and set 1/1000 @ f2.5 and ALL were over -exposed -- I also took a series with my Canon Speedlite 199A set to 'auto' first then set to 'manual 1/60 ' on shutter and they ALL were the SAME density OK -- just the daylight ones are over-exposed .
It look as if the faster speeds are running very slow as one I took on the Kodak T-max 400 with the Canon 200mm f4 SSC lens was over-exposed but also had camera-shake although I had set 1/1000 sec speed.

You would think that running an f4 @ 1/1000 would be underexposed. Was the de-squeak done with the proper "clock oil" ? Maybe just shoot the heck out of it without film you may have the problem of not being used enough regularly. I know several times I get a camera and the only problem was under usage over the years. Shoot a half dozen rolls put my A1's in fine shape again. Good luck.

I used 'Three-in'One oil as I had not heard of 'Clock oil' but I used a very small amount -- had no 'syringe with needle' so dripped down one drop off end of a piece of electrical fuse-wire in the base-plate space as shown on many videos.

You would not think that such little oil would do much to disable the camera. Go with the shooting the crap out of it with no film and hope for the best. Use different lenses and settings and shoot away.

I had a look after taking off the base plate and took off the base plate of an AE-1 to compare-- I saw a small 'lever' on the side where I had poked down some wire with oil on it to lubricate for the ' Canon Squeak' which was UNDER a small part whereas on the AE-1 it was IN LINE -- so I touched it gently and it came 'in line' like the AE-1 part and on firing shutter stayed that way and on looking through the back at shutter 1/1000th seems much faster -- now to do another FILM test !! ----

GOOD NEWS !!! I did a series of test shots using 'P' 'AV' and TV' -- I used a Weston Master III meter to get an Incident reading to see if Camera meter agreed -- it said 1/1000 @ f2.8 for test scene which agreed with Camera meter -- so I did a series from 1/1000 down to 1/30 and on processing all were very close in density so the 1/1000 and 1/500 which were giving bad over-exposure no longer do it and it looks as if moving that little lever in line has done it !

Here is a photo showing the straight 'lever' that was out of line :
Canon A-1 test by Peter Elgar, on Flickr

I probably moved that unintentionally while probing my A1 in squeak removal process but I have never had the problem that you described.