I know you are getting answers from Realtors out of Long Island...business is done differently elsewhere...
Where does the seller get the money to pay the realty fees?
Is it possible that a real estate attorney would know just as much as an agent?
Folks, the real estate agents have no shame in asserting patent falsehoods as truth. Dump the agents, pick up a competent realty attorney, and do the deal.
It is unbelievable that adults still consult real estate agents and believe what politicians say.
Another strategy is to contact the listing agent of the first house and see if a quicker response is possible. You could also feel out the first listing agent see if you are in the ball park on price and try to get a sense of whether the seller of the first offer will accept it or counter at close to your offering price.
A third thing to try is research activity on the second home and ask the agent for that listing how much activity they've experienced and whether they anticipate multiple offers. You could also ask straight up what kind of price the seller is expecting, to make sure the anticipated selling price isn't out of your range.
Finally, think about why you like the second home more. If it's about location and schools, then I can understand why you would like it more. If it's about condition and the first house can be brought up to par to the second one, and be had at a more reasonable price, you might want to stick with the first one if you think it's a surer bet.