Yes, you can pray Tahajjud without having to go to sleep.
The classification of night prayer is based on human understanding, not a strict requirement set by the Prophet Muhammad (peace be upon him).
وَمِنَ ٱلَّيۡلِ فَتَهَجَّدۡ بِهِۦ نَافِلَةٗ لَّكَ عَسَىٰٓ أَن يَبۡعَثَكَ رَبُّكَ مَقَامٗا مَّحۡمُودٗا
And from [part of] the night, pray with it as additional [worship] for you; it is expected that your Lord will resurrect you to a praised station.
Surah Al-Isra (17:79)
Here it mentions that any part of the night can be considered the time for night prayer, including Tahajjud.
In the last 10 nights of Ramadan, it is encouraged to engage in more worship, and some scholars recommend staying awake until the Fajr prayer, especially on Laylat al-Qadr (the Night of Decree).
And Allah knows best