If not for the other, now there are many good (and not so expensive) phrase books out there in all kinds of languages and when I was in Spain for a week about (ups, so long ago?) 23 years ago I had no particular clue of Spanish but yet I've tried to learn some. Comming there at first I was more or less quiet but the more the week went toward the end the more I babbled in Spanish, tiny little things like ordering sangria and beer, having my coffee, saying "hola", etc. You, with so many traveling experiences probably can picture the expression on local's faces when they had heard their language spoken by the tourist who could very easily resort to English. I felt I made much more with my trying to speak then just speak, I showed the respect toward them, their language and country. And, yes, of course, if I'd start to think how bad my Spanish was, I probably wouldn't even open my mouth. So, I believe, if even just "hello" in whatever language, it makes wonders. But, yes, there are "bad" people allover the World who don't charrish the effort of either, being local or tourist. And I know very well how it goes with nations living together in one country ... we had many experience with that ... 6 nations, more languages ... and we weren't always like "brothers" with each-others either regarding culture, language usage and so on. So, yes, I know the point.
But, to return to the topic itself (speaking about Welsh), I sometimes wonder if we, who are not Welsh, can really contribute something to things to get better in Wales itself. The language will live this way and grow, but will it contribute to the state of mind of the people? I sincerely hope it would in a way. And you do just the right thing by my opinion with speaking some Welsh to your students. In time at least someone could show interest in the language and eventually start to learn it. You never know unless you do things which could potentially lead toward that goal.