I'll split my answer into two:
Are you speaking the language?
From the spell description:
This spell grants the creature you touch the ability to understand any spoken language it hears. Moreover, when the target speaks, any creature that knows at least one language can hear the target and understands what it says.
As it is written, I understand it as: You are able to hear the language and people hearing you understand as if you were speaking a language that they know. It is not clear, though, which language is being heard, mainly in the case where the listener knows more than one language.
So, in your example, let's say there is this Dwarf City where everyone speaks Dwarven (and only dwarven, anyone not speaking dwarven is seen as a barbarian and should be killed) and you want to blend in. Nothing on the rules state that the language they will be hearing would be Dwarven instead of simply Common.
Is it obvious that you are using a spell?
The casting takes one action and the duration is one hour. You don't have to do anything in order to maintain the spell, so it shouldn't be obvious that you are using a spell, at least nothing in the text describes it.
I think I've been using this phrase too much lately, but... ask your DM. If you are the DM, decide on what makes more sense. RAW, it doesn't seem to explicitly prevent or allow what you want to do.
Tongues is more of a flavor spell (unless your DM wants you to use it and creates a scenario where you have to communicate with something or someone that doesn't speak any common language with anyone in the party) so actually giving it an interesting utility should be fine and the player using it could feel motivated.