I've Got This One!

So I've had a number of first, sometimes second dates lately. The issue of who is going to pay the bill is always awkward, but on a first date it's even worse! My solution is to sit there and ignore the bill - pretend like it was never put down, just keep talking and being my sweet, charming self. I always make sure to thank them for the dinner...but I'm the one being courted here! Having me pull out my wallet is not the way to woo me!
According to one party, I should always at least offer. I should say "I've got this one" and give them the opportunity to say "No, I'll get it." I feel like by using a statement they don't have an opportunity to say no. I should use a question. "Do you want me to get this?" But H said no. I need to offer, not ask. Perhaps this is a Midwest thing?
Another party said that I shouldn't even offer to pay - she's the one saying they were trying to woo me! Allow myself to be woo-ed. She said that I shouldn't acknowledge the bill because I shouldn't be expected to pay.
I tend to agree with the second (perhaps smarter??) party. Plus, if I'm trying to check my options out and date numerous guys...I can't afford to get stuck with all those bills!
I'm totally for splitting costs of a relationship - once it is a relationship. Right now, I'm still testing the waters! What should I do? Offer to pay? Keep on keeping on? I don't want to come off as a spoiled brat (no comments here, please) but I also don't want to go broke trying to date! Also, if I should offer to pay (and expect them to say no), how do I do it in a way to not encourage them to say ok? Help!!