Yes! (But in some communities, it may be discouraged or even "not done.")
But doesn't a name need to be official before you can use it?
Nope. That's true both under the secular law and Jewish law. Well, in the sense that neither prohibits you from calling yourself whatever you want whenever you want. Want to call yourself Beeblebop MacShoe? No one will stop you except to help you put on that straightjacket.
Should you use your Hebrew name before conversion? Or ever? Maybe, maybe not.
It's unlikely that a rabbi would ever preemptively tell you not to start calling yourself by a Hebrew name, but you may get a talking to after the deed is done, if the rabbi disagrees. If conflict makes you uncomfortable, it would be best to get a green-light to use a Hebrew name pre-conversion. If anyone other than your sponsoring rabbi or beit din rabbis tries to tell you you "can't" do it, ignore them. They're probably just yentas. Be thankful when yentas and haters out themselves. It's much easier to ignore them going forward.
Remember that you are not, nor should you ever be, required to use only your Hebrew name as your name on an everyday basis. It's your name, and you're free to use it as frequently or as infrequently as you like. While most converts do use their Hebrew names in some way, it is not required. (Of course, in some communities, not using a Hebrew name may be seen as a kind of rebellion or as an unwillingness to fully join the community. Even there, it won't be a "rule," it's just "not done." Not my cup of tea, but maybe it's yours.)
In almost all cases, you will choose your own name. It's an intensely personal decision and can take a long time or be blindingly obvious. Only in a very few communities will a rabbi choose your name for you, and those communities are usually Chassidic (though probably only a small percentage of Chassidic communities). Personally, I'm suspicious of situations where one person is given that much power over his followers' lives, and I would view that as a serious red flag. Others disagree with me and feel this is superholy and the name is drawn down from heaven on your behalf. I don't think any rabbi on this earth today has saintly superpowers ("ruach hakodesh"), but as always, others disagree.
As I say in that blog post, there's a difference between "Your Hebrew name is going to be Y." and "Have you considered the name Z? I think it might be a good fit for you and your personality." One is likely to be seen as forceful and intimidating, and one sounds really thoughtful.
Your English and Hebrew names are both you, and probably one will speak more to you on a daily basis. For me, that's my English name. However, I am a rare person who converted and still uses her (obviously) English name. People try to say that it's not a "Jewish name" to me, but neither was Alexander or several other names until it became popular. In my case, my name isn't commonly used even in the secular world, but I have a great comeback: "It's very popular in the London frum community, actually." (Or so I've been told. It has since caught on in Northern NJ too.)
Most diaspora Jews have a "secular" name on the paperwork, and many use those names at work. In fact, it's very confusing to do business with people I know in the community because I often have to learn a new name so I can get the secretary to transfer the call to the right person! I tell you this so that you'll realize it's not an all-or-nothing decision, whether you're a convert or BT or frum-from-birth. Almost every person in the community has to deal with this issue at one point or another. Often, we confront these questions multiple times in our lives.
And...you can always change your mind. I happen to love my English name and am very attached to it, but I chose a Hebrew name I could see myself using if I lived in Israel. People who don't speak English natively tend to mangle my name, and I'm unreasonably bothered by that, so I probably wouldn't use my English name in Israel. Our self-definition is malleable and always open to re-definition.
So let's cover some potential scenarios:
You can be Chaim in shul and Greg in the office.
You can be Chava to everyone new and Elizabeth to everyone who already knows you (unless they want to call you Chava too).
You can be Sarah Leah to everyone else, but always Kara to your mom or other family members.
You can be Ezra to everyone, but still Mark to the cashier at the liquor store who checks your ID.
You might be Erica on your job application, then ask everyone to call you Ilana instead when you get the job.
You can be Kochava online, and Skylar in the real world.
And you'll always be "Mr. Ackertonson?" to the telemarketers.
Life's funny, and people are weird. Embrace it or you will become very bitter.
If you do choose to go by your Hebrew name in one or more of these contexts, you'll always have someone who gets it wrong or refuses to change over. You have to consider whether you can live with it, whether the battle is worth it if not, or whether this person needs to be in your life at all.
But should you really do all this before conversion? My only caution is that once you start going by a name, you should stick with it. It gets too confusing, and it hurts our games of Jewish geography if you change names a couple of times. People might start getting Judgey McJudgerson about it and wonder if you're right in the head or noncommittal about this conversion business. You might start looking like a poser, and everyone hates posers.
If you're not sure about your name (remember that it's a lifelong commitment), don't use it yet. Ask your friends whether it fits you, ask your family how it sounds, say it out loud in the mirror a few times, but don't start putting it on job applications.
You can even ignore the question of a name altogether until it gets closer to your conversion. Or only delay the decision on whether to use it or not. Procrastination is ok, and the beit din is unlikely to ever ask you how you plan to use your Hebrew name. On the other hand, they might ask you those questions if you start asking people to call you by your Hebrew name.
A note on last names: unless you intend to legally change your last name, I don't suggest using a "Jew-y" last name just to "fit in" better. If you get caught, people will think that's weird and suspicious. You might even be viewed as a security threat.
Once you've converted, you're pretty much stuck with the name, so you can use it or not as you like. Before then: kick the tires, but don't commit until you're ready.
More about Hebrew names and legally changing your name can be found on the Hebrew Names page!
So... when did you start using your Hebrew name, and why then? If not, why not?