Can we: Travel to the US with the current passport.
It depends on what is the country of your passports. The U.S. has a requirement that entering nonimmigrants have a passport valid for 6 months after the end of the intended duration of stay. That means they will give you a duration of stay that ends no later than 6 months before your passport expires, and if your passport is less than 6 months away from expiring, that means you cannot enter.
However, the U.S. has agreements with many countries where those countries agree to accept their passport for return for up to 6 months after the passport expires. For passport holders of these countries, the U.S. will give a duration of stay that ends no later than the passport expires, and you can enter as long as the passport is not expired. For a list of these countries, see here.
Note that even if you can enter, if your passport is close to expiring, the duration of stay on your I-94 is likely to be limited by the passport expiration, and not go for the full duration of the H1b petition. In that case, after you renew your passport (in the U.S.), you should get CBP to adjust the date on your I-94 so you don't fall out of status.
Can we: Renew the passport but show the visa in the old one.
Yes, as long as both passports are from the same country and the same type of passport (i.e. both ordinary passports, or both official passports, etc.).
Can we: Renew passport and try to get the H4 stamped before we travel (really not sure we have time).
There is no need, since your existing visa is not expired.