Oh my, you don't realise it, but you're doing it right, hah!!
You see, in the lower register, one can only go so low, especially a female. And everybody has their limits. And when you reach your limit in the lower register with "regular" singing, you have to pretty much speak-sing, otherwise you'll probably directly go into a vocal fry, or nothing may come out at all because it's simply not possible for you. It does sound kinda weird, but it's right.
Then again, you can only go so deep that way, too, before you hit the vocal fry, or your limit.
It sounds to me like you just have a fairly high voice, though you want a deeper one... It is possible to extend the range of your lower notes, but it's usually much harder. All you can do is practise each day, using your modal register to speak-sing and work on increasing the volume and/or range of those lower notes. Like Tenez said, do scales every day and work on it like that. But - and this is with everybody - low notes honestly aren't meant to be belted, and it's natural for somebody's very high or low notes to even be close to a whisper.
Also, just be careful with this. Trying too hard to extend your lower notes or singing too much in a range of notes you aren't comfortable with, or by practising the wrong way, will not only risk damaging your vocal chords a bit, but it can also shorten your range of higher notes.
My advise to you would be to actually embrace your higher voice and work on the higher notes.
And if you're worried about not being able to sing certain things - like those Ayumi songs - learn to adjust your voice accordingly. This could mean changing your pitch to match your voice with the song, or it could mean completely singing in another register.
I know, it can be very frustrating whenever your notes don't match up with what you want them to... I actually have the complete opposite of you. Since my voice is very deep, the verses in Duty, for example, are terribly easy for me, but then I have to focus for the end of the chorus.
About the tightening of your throat.... Does this happen with only the low notes, or does it happen all the time? If it's just with the low notes, that's natural. It means you've reached your limit (at least in that register) and need to take it kinda easy. But if your throat often tightens up, no matter how high or how low, you're probably singing wrong. D: