哪一個 My BlackBerry 9810 has not only a larger screen but also a keyboard. 或 The 9810 model not only has a larger screen, but it also has a keyboard. 聽起來更自然？

I think both are natural.
両方が自然です。

The one where you say "My BlackBerry" would be okay for with friends or if you're talking about something you own.
「My BlackBerry」の方は友達や親しい人と話している時や自分の物について話している時に用いる言い方です。

The one where you say "The 9810 model" sounds better for an announcement or presentation. It sounds a bit more "official".
「The 9810 model」の方は、発表やプレゼンなどに用いる言い方ですね。公式的な印象がします。