Should we try to outbid another house buyer whose offer has been verbally accepted?

My husband and I are trying to buy a house in a highly competitive real estate market. Recently we made an offer, which was accepted by the seller, on a home we loved. But before the contract was signed, another buyer came in with a cash offer higher than ours — and the seller dumped us and accepted the higher price! We were blindsided, and discouraged to say the least.

Now we’ve seen another house we like a lot; our real estate agent tells us the seller has accepted an offer, but she also implied that if we can do better, the seller might consider our bid. What should we do?

— Frenzied