Can I pray before the adhan?

Yes, if the prayer time has already started. The adhan announces the time; it isn't what makes your prayer valid. What you can't do is pray before the time begins: a prayer prayed too early doesn't count and must be repeated once the time comes in.

Time is the condition, not the adhan

Every obligatory prayer has a set window, and entering that window is a condition of the prayer:

أَقِمِ ٱلصَّلَوٰةَ لِدُلُوكِ ٱلشَّمْسِ إِلَىٰ غَسَقِ ٱلَّيْلِ وَقُرْءَانَ ٱلْفَجْرِ "Establish prayer at the decline of the sun [from its meridian] until the darkness of the night and [also] the Quran of dawn." Quran 17:78

The adhan is a sunnah (some scholars say a communal duty) that tells people the time has come. The Prophet ﷺ said that when the prayer time comes, one of you should call the adhan, and then you should pray as you have seen him pray (Sahih al-Bukhari 631). So if your app says Dhuhr began at 12:52 and your local mosque calls the adhan at 13:05, you may pray at home at 12:55.

Why the adhan and the time can differ

Mosques sometimes delay the adhan by a few minutes for safety or to gather people.
Different calculation methods give slightly different times, especially for Fajr and Isha. See Can I trust prayer time apps?
Some mosques have two adhans for Fajr. In the Prophet's ﷺ time, Bilal called an early adhan at night to wake people, and a second caller announced true dawn. Don't pray Fajr after the first one if it is before dawn.
Many Western cities don't allow a loud adhan, so you rely on the timetable anyway.

Praying at the mosque

If you are at the mosque, wait for the congregation. Use the time between the adhan and the iqamah for a sunnah prayer: the Prophet ﷺ said there is a prayer between every two calls (Sahih al-Bukhari 624). Praying the obligatory prayer by yourself just before the imam starts, in the same mosque, isn't good manners. See How many times a day do Muslims pray?

Should I say the adhan myself?

If you pray alone at home, at work or outdoors, saying the adhan and iqamah quietly is a sunnah, and the iqamah alone is a good minimum. It isn't required, and a whispered "Allahu Akbar" to begin is enough if you're in an office. See How do I pray at school or university?

If you prayed too early

Mistakes happen, especially in the first months. If you realise you prayed before the time began, simply pray again once it starts. There's no sin for an honest mistake.

Key points

You can pray once the time starts, even if you hear no adhan.
A prayer before its time is not valid and must be repeated.
At the mosque, wait and pray with the congregation.
Saying the adhan or iqamah yourself is a sunnah, not a must.

Tip: In your prayer app, set the notification to the start time and add a two-minute "cushion" before praying. It covers small differences between apps and keeps you safely inside the window.