I'll tell you the sad story, but some solutions do exist.
Unfortunately, this version of Delphi had limited support of Unicode. Borland VCL was not Unicode-enabled at all, so, with the controls you are using, you cannot do anything at all. You would need some custom controls replacing VCL.
In 2005, I found a relatively smooth way to derive VCL controls to add Unicode support. I though that rewriting the VCL completely would be infeasible, even though any other way though to be impossible, as far as I could see from Delphi community material. I found a tricky way: I inherited select controls from VCL controls and develop a trick based on Windows API: re-registering of the window class on the fly, adding Unicode support and keeping everything else. This is the idea you could use.
I published this work in the printed The Delphi Magazine
S. A. Kryukov, ‘Unicode Controls: What Can VCL Do?’
, The Delphi Magazine, 116 (April 2005), p. 33-43
Unfortunately, pretty soon after that this magazine was closed. I wasn't a subscriber of this magazine myself. I don't have a hard copy, only a copy of my original work, before editor's proof. The on-line access to the code was removed with time, but I have it. I used to see some (illegal) copies of the code of this work on the Web. Personally, I would not mind if anyone uses it. And I have a very little incentive to re-print it (even assuming it would be legally fine), as that version of VCL is quite obsolete.
So, what else can I advise?
First of all, you can use later version of Delphi, starting from Embarcadero Delphi 2009 (Delphi 12, code named Tiburón). This is when full Unicode support for VCL was first introduced: http://en.wikipedia.org/wiki/Embarcadero_Delphi
Another good option would be switching to open-source Free Pascal:
One great benefit of Free Pascal is that it is truly multi-platform. And it has Delphi compatibility mode. Unicode is of course supported.