I'm a non-native speaker of Russian, and my given name is Andrew. Is it more normal to call myself "Эндрю", which according to Wiktionary is the transliteration of my name, or "Андрей", which is a cognate (ie an etymologically related word)?
If you call yourself "Андрей" you'll have tons of questions from everybody whether you have any Russian ancestry. It's typical for non-native speakers of Russian to call themselves with their real names, "Эндрю" in your case. Still, I knew a Vietnamese guy named Ha Hai Tan who called himself "Толя" (it's short from "Анатолий"). When I once asked him why he chose "Толя", he said, "Well, Tan – Толя, they sound very similar." ))
Well, call yourself with your real name "Эндрю" if you'd like to avoid such questions and surprised looks, or choose "Андрей" if you want them, or if you decided to convert to Russianhood.
And actually, "Эндрю" is not a transliteration of "Andrew", it is a transcription with the Russian letters.
I would use the name you are most comfortable with. Andrew is easy to pronounce (as opposed to, say, Jügderdemidiin Gürragchaa), so the majority of Russians won't have a problem with it. Your name is part of your identity and I don't see, why you should adapt it.