The problem with being "obsessed" with grammar is not that it's a waste of time learning the details, but that you become so afraid of making a grammar mistake that you're shy about speaking and writing Russian. But look at it this way: If an ESL speaker says "I'm going at supermarket" -- instead of "to the supermarket" -- absolutely every native speaker of English will understand the meaning, in spite of the mistakes. (And English speakers who've studied other languages will be very sympathetic, because they know from experience that little words like prepositions can be notoriously difficult.)
Still, you obviously need SOME grammar, such as the rules for forming the tenses of regular verbs. (You'll still screw up the irregular verbs, but guess what -- young Russian kids do that, too, just like English-speaking children say "writed" and "buyed", but adults will generally know what the kids are trying to say.) So, in the beginning, there's no getting around a certain amount of rote memorization and recitation of past and present verbs, or the six case endings of "type 2" feminine nouns, or the conjugations of easily confusable verbs like "lie" and "lay" or "sit" and "set" (which, alas, are just as much of a headache in Russian as in English, if not worse!).
The good news, on the other hand, is that once you've absorbed some of the basic patterns, you'll no longer have to worry about rote-memorizing 12 different forms (i.e., six cases, singular and plural) for every new noun you learn. You'll just go, "aha, this noun is masculine and it behaves like the masculine nouns I've already learned."