I was thinking of Gatorskin's for commuting
I cannot recommend Gatorskins. They have a horrible rolling resistance for a road tire. Besides, despite their claimed puncture protection, they puncture early and often. In fact, Continental claims the Grand Prix 5000 to have an even better (supposed) puncture protection so if you are one of the folks who think puncture protection is useful (hint: it's not, it only increases rolling resistance), you aren't getting the "best" by buying GatorSkin. Instead, select Grand Prix 5000.
If you want to compare the rolling resistance of Continental UltraGatorSkin with Continental Grand Prix 5000, see this: How to estimate the speed of cyclist with given parameters
have a Giant hybrid bike, that currently has 700x38 tyres on 622x29 rims.
Is this a typo or is the rim actually 622x19?
622x29 is very wide no matter whether the 29mm is the external or internal width. You won't be using 32mm tires on 29mm rims. However, 622x19 rims do support every reasonable tire size from 28mm up to about 45mm.
I need to replace the tyres, as they are getting old and was wondering if I could change the tyres to 700x32, do you think that it is a good idea, if not any other suggestions?
The 700x32 (or 622-32) is about the best possible size you can pick. The reason for this is as follows: high performance tires like Continental Grand Prix 5000 are not available for wider sizes than 32mm. On the other hand, within some certain sane limits, you want to prefer wider tires over narrower tires because they have lower rolling resistance and higher ride comfort. The only things you lose when selecting a wide tire is weight (lose 0.1 kg of body weight and the difference goes away) and air resistance (ride few hundred meters more on the drops per 10 km of distance and the difference goes away).
but am a bit worried about how they will handle in the wet.
If you are worried about the relatively smooth tread pattern, you'd be wrong. Car tires hydroplane because they are smooth and wide and cars drive at high speeds. Because of this, car tires have a tread pattern. Bicycle tires are curved and narrow and bicycles ride at low speeds, so bicycle tires cannot hydroplane. Thus, the best bicycle tires have a completely smooth tread, no pattern.
Oops! You cannot sell a smooth treaded slick tire because most bicyclists have seen car tires that have a tread pattern and thus are afraid of riding on slick bike tires. Thus, manufacturers like Continental put ridiculous "micro patterns" on tires like GP 5000 and UltraGatorSkin. These micro patterns serve no useful purpose.
The best tires for bicycling on the road are slick tires. The second best tires are 99% slick tires like GP 5000. The tread pattern reduces the wear life of the tire a bit, makes it a bit more expensive to manufacture, and reduces traction a bit. But only a bit, so little that it doesn't matter.