You do have a dilemma. I can't tell from your comment if it is a matter of just not wanting to take your regulator out of your mouth at all while underwater, or not being able to take in / spit out seawater afterward. If the former, your options are more limited. If the latter, I usually don't find rinsing my mouth with
seawater is particularly helpful - although I will do that from time to time - because I end up being more 'thirsty' / have an even drier mouth afterward. In any event, if you do use a u/w mouth rinse with sea water, definitely don't swallow. I don't find that rinsing my mouth with it causes a problem, but swallowing it can certainly cause emesis.
As several have noted, work on increasing your comfort with removing and replacing your second stage underwater. That is an essential / core skill which should become second nature. Once that skill is mastered you have several hydration options during a dive. For example:
1. Deep Sea Supply markets a small (14oz) hydration bladder (
DeepSeaSupply - Product Detail). Although it is designed and marketed specifically for use with back-mount doubles, it can fairly easily be adapted for other use (e.g. simply putting it in the pocket of a BCD, and pulling out the tube whenever you want a drink).
2. Some divers even carry a MSR or Camelbak (
http://shop.camelbak.com/tahoe-lr-/d/1317_cl_4671)` hydration bladder, with a bite mouthpiece, during dives. I think these are a bit bulky (even the 'smallest' MSR Hydronedary is 2 liters), but they can be used underwater.
3. Some divers like to use drinks such as Capri Sun (
http://www.caprisun.com/), with small straws. You can also get some small children's juice / liquid containers with 'closeable' caps, and refill them with water.
I suspect one of the reasons that the Deep Sea Supply product was developed for doubles is that it is more likely for dry mouth to become a problem on longer dives. But, there are times during even a 50-60 minute recreational dive that I notice my mouth becomes dry. Usually, I find this to reflect either my overall state of hydration, or my diet immediately before the dive.
I have not personally observed that the particular mouthpiece I am using makes a noticeable difference in the likelihood of developing dry mouth. But, I now use only Apeks Comfo-Bite mouthpieces, and at least one poster suggest that this mouthpiece is less likely to produce the problem. If you don't own your own equipment, mouthpiece selection is for the most part a moot point, although some diver's do carry their 'personal' mouthpiece, and some tie-wraps with them, and swap out the mouthpiece on rental equipment before the dive, then replace the original mouthpiece before returning the rental unit.