There were a couple of threads on logic level voltage conversion recently, so you may want to search then out. Also, there is this doc:
**broken link removed**
Basically, the resistor method is only useful occasionally. The reason is that you now cannot supply much current, so it will take longer to
charge the capacitance at the input of the circuit, so your waveform doesn't look square at high frequencies. The way around this is to
use a dedicated logic converter ic, or to use a switching device (e.g. MOSFET) to do the level conversion. Anyway, the doc I mentioned
describes it a lot better than I can. But you might be ok, and the resistor method may be ok.
Also, you're right, your issue is due to the different grounds that you have got, I couldn't see why you did not connect the two
grounds together. When your TX pin goes high or low, it is quite undefined what voltage level the FTDI RX pin will see, since the common ground
reference is not connected.
For example: Lets say your Hardware RX pin is at 3.3V. That means that the reference GND that the FTDI chip 'sees' is that 3.3V through the
lower resistor, i.e. 3.3V. So, if your Hardware TX pin is at 3.3V, then it looks equivalent to GND to the FTDI chip. If the Hardware TX pin went
to 0V, then the FTDI chip would 'see' -3.3V with reference to it's GND.
I think you should confirm your DC converter is generating 3.3V, and that your Hardware TX/RX pins don't seem to have anything more than 3.3V
on them, and then you can quite safely connect the two grounds together.
If you're worried, stick a low value resistor temporarily in series with the supply and your circuit, to limit the maximum current that can be drawn,
while you're testing.