After all, the p-value will automatically decrease when N increases.
This statement is not correct. If the data continues to come in the way it has, then the p-value will decrease, but if the weak correlation is due to random noise, then more data will make the p-value increase. Imagine if you flip a coin 20 times and get 14 heads. That's unusual since 94% of the time you would get less than 14 heads, but not outside the realm of possibility for a fair coin. If you flip the fair coin forty times, it is unlikely you would end up with 28 heads total. If you did, you would absolutely say the coin is not fair.
Your statement about p-values between .05 and .10 is equivalent to saying "I got 14 heads out of 20 flips therefore the coin is not honest and I believe that if you continue to flip, you will see that." Could be, but it also could be you are just responding to statistical variation.
I was taught that if you find strong effects on a small N that are close to being significant, you can assume the effect is actually there.
I disagree strongly with this statement. If you flip a coin four times and get four heads, would you say the coin is unfair? That happens occasionally just by random chance. You need to collect more data.