There is no Quranic verse or any Hadith saying don't keep the engagement. You can keep engagement but it's not the part of Sunnah of Prophet Muhammad (pbuh). It's Mubah (optional) but remember it shouldn't break any Islamic law. Now coming to the basic question that how should a boy and girl behave after they decide to get married maybe a few months later. Please note that your behave with your future husband or future wife would not be the same as wife and husband please remember it and that's a western culture once you are engaged means no problem you can do everything even before marriage you can do everything and after engagement you can do everything and after marriage also everything. That is a western culture.
Here in Islam your Hijab should be maintained and as far as the Quran is concerned its mentioned in Quran in (Surah Al Noor: Chapter 24 Verse 30) saying to men and women to lower their gaze that can relax because engagement has already taken place but relax doesn't mean the complete freedom but yet Hijab should be maintained same as with your Non Mahram. Please remember he is yet your Non Mahram. Fine if during the first interview if the husband insists that you can relax your Hijab a little bit because i want to have a better look at you so certain Hijab can be relaxed not fully. Fine he wants to have a better look at you can remove a scarf but it doesn't mean complete Hijab should be removed and that also doesn't mean always otherwise Hijab should be maintained just like as Non Mahram.
If you have to meet anytime for a discussion but remember Mahram should be there either the bride's brother is there or the father or the uncle or would be bridegrooms mother or sister. You can't say ok now we are engaged let's go for a movie together. You can't go out together in the garden alone. No not at all alone because yet you both are Non Mahram. If it's important you can have a meeting in the house with the people which I have mentioned above but don't unnecessarily keep on the meeting. We are going to get married in six months so we will meet twice in a week. It doesn't make sense at all.
It's better for you to do Nikkah don't go for marriage yet than meet every day no problem. If you are saying I am delaying the Nikkah because I don't have a house. If you are not prepared and you want to shift. No problem do the Nikkah. Simple Nikkah and yet a girl can stay with her parents and a boy can stay with his parents. If you meet every day no problem and then all the Hijab can also be relaxed but remember before Nikkah you can meet but with some guidelines. Our beloved Prophet Muhammad (pbuh) said if two Non Mahram are in a room the third person is a devil so if you are engaged and you are alone in a room its totally Haram in Islam.
If you want to speak on phone once or when it's important you can. You can speak but don't make it a habit regularly and you can speak once a week or once in three days. You can but note that the talk is more about Islamic and within Islamic guidelines rather than laughing and joking and remember the way you talk is a part of a Hijab... Hijab is not only about clothing, Its also about the way you talk, the way you behave. You can meet to discuss certain issues of marriage but with your Mahram but don't relax it too much because maintaining the Hijab and modesty is important.