I'm having the same issue with my 2001. My fob asks for a 2035 battery so that's what I kept using and it would work intermittently but well enough for a week tops. Last month I accidentally shoved a 2032 battery in it because that's what my other vehicles fob takes. Now its been working way better for over a month! The lock button works great, the unlock works only after I hit lock once and then unlock. It's a massive improvement with the higher capacity and slightly thicker battery. Worth a shot!
Also, does anyone know where the antenna is for the keyless? I want to manually inspect it but can't find it's location in my Haynes manual or by googling.