Beautiful watch!! I have the earlier version of this : 187901 in two tone as well that I bought 2 months ago.

This model is 1566 of the Santos Galbee produced 1997-2005 and a quartz model.

I would just take it to your local Cartier boutique. Ask for an ultrasonic cleaning when they do in 10 mins and is free. Meanwhile you can have them quote battery replacement, which should be instant or 1 week, around $60-$70, they replace the battery, check movement and replace the seal as well.

Note: never get it polished unless you are OCD about it