Is it alright if an engaged man before sleeping thinks about romance with his future wife (not about sex)?

Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

These kinds of imaginings are but insinuations from Satan. Satan does not spare any effort to induce the person in sin. Imagination is a first step, the second step is making contract then meeting, and so on and so forth. Allah says: "O you who believe! Follow not the footsteps of Shaitân (Satan). And whosoever follows the footsteps of Shaitân (Satan), then, verily he commands Al-Fahshâ' [i.e. to commit indecency (illegal sexual intercourse, etc.)], and Al-Munkar [disbelief and polytheism (i.e. to do evil and wicked deeds; to speak or to do what is forbidden in Islâm, etc.)]." (An-Nur 24:21)

So, getting pleasure in such imaginations might lead to committing Haram and fulfilling one's desires in a forbidden way like masturbation.

As a result you should refrain from elaborating on such imaginations. They expose you to great dangers and make you fill your time with nonsense.

The Muslim should read when going to bed the Adhkar related to sleeping like reading Al-Kursi verse, the last verses of the surah al Baqarah, the three last Surahs of the Qur'an, saying Subhana Allah 33 times, Al Hamdulillah 33 times and Allahu Akbar 34 times.

Know also that the woman to whom you are engaged is alien to you until the marriage contract is established. So you should not stay in seclusion with her, shake hands with her or talk to her in a way that arouses sexual desire.

Allah knows best.