The central timer module is usually just a function of the GEM module which is behind and to the left of the radio.
Edit: The generic electronic module (GEM) is equipped on four-wheel drive vehicles, or two-wheel drive vehicles with power windows. On these vehicles the GEM incorporates the CTM function so no separate CTM is needed.
Two wheel drive vehicles without power windows do not have a GEM and then have a CTM module instead.
The GEM is how it does the delayed modes slower than low speed, but a failure of the GEM would tend to make them not work at all, unless you had a windshield or roof leak that got condensation into it to create a short.
What is the rate of occurrence of the problem and the duration? Wipers always on? Wipers randomly come on a single or how many wipes? Does the washer spray?
As far as aftermarket switches, some people say they're fine, but if I can restore a factory part to like new function I'll always go that route. When my MFS went bad (manifested as an erratic turn signal) it had no real wear in it, just corrosion of the copper contacts so I opted to open and clean it out (fine sandpaper or steel wool or whatever (metal polish, etc) will work, make sure it's completely cleaned of debris), then put dielectric grease on the contacts.
I would not only return the new one if you have the option, but also take the old one apart while you have it out and clean the contacts so you don't have to pull it again to do that later. Just lay it flat on a table to take the screws out, there is one bit on the end that can pop out but it's otherwise mechanically simple inside.
That doesn't address the switch mechanism in it for the wipers though, that is accessed through the end cap on the lever. The pin and spring in it can come flying out so you have to be careful not to lose them. More info here:
Intermittent (Phantom) Windshield Wiper Fix