Some girls like getting texts every day, definitely, and I think you did a sensible thing by holding back on the texting to see how she took it. Her reaction of asking you why you haven't texted in ages is a good one, it shows she's been thinking about you and likes receiving your texts - reply to it and make a joke of it, you could even be honest and say you didn't want to come across as being "too forward" - don't use the word "desperate", it makes it sound like she's your only option and she'd take that offensively - and her reply could give you an indication of how into you she is. Bottom line, don't be too serious, it's early days, be jokey about it but without being a jerk.
It's generally a good sign if she teases you, assuming you mean in a flirty way and not a patronising "awww, aren't you cute" way like she might tease a younger sibling. While I'd personally be a huge snob about girls who use "lawlz" in anything but an ironic manner, things like "hahaha" are a good sign too, she's telling you she finds you funny and she wouldn't put it if she didn't, there'd be no point.
Just try not to think too hard about it, like I said it's early days, just go with it and carry on texting as much as you want to; she can simply not reply if she thinks it's too much :)
· 6 years ago