@zurlita
I don't think there's any way to get public insurance until you're in the social security system, and you can't be in this until you're legally resident in Spain (or, officially, as you have Romanian passports, EU Citizen Registration or Certificado de Registro de Ciudadano de la Unión Europea, which I guess is what you're referring to as a CUE).
https://inclusion.seg-social.es/web/migraciones/w/certificado-de-registro-de-ciudadano-de-la-union-europea-hi-101-
You can get a NIE independent of being a resident (or you can get your NIE issued when you do your registration step above). But I'm pretty sure that the NIE isn't sufficient to get you the insurance you're discussing (you'll need your registration certificate).
The requirements for registration are proof of funds and proof of health insurance (as you know).
I'd be surprised if ALL Spanish health insurers would decline to give you a policy, especially if you don't mind a high premium.
Apart from that, as you're OK, YOU could apply for your registration (as no problem with insurance), then investigate the public option for you + spouse (and maybe it's helpful to do a family reunification instead of a 2nd registration for your husband).
There is also the sneaky option of buying a very expensive policy (so you have cover), but making sure there is a cooling-off period (usually there is), or there is some kind of cancellation option. Then you can use it to register, and once you have your certificate you immediately cancel the policy and look for a cheaper option (and as you're then a resident, you can pursue the public options).
And it doesn't need to be a Spanish policy. If you've been in the USA for 19 years, I'm sure you've had plentiful health insurance there, so I'd guess there might be an overseas option available to you. The requirements are that it's proper medical insurance (not travel insurance), that it covers Spain, that it has a certain minimum level of cover (you'd need to check, but I think it's small, maybe 60k-100k euros), and ZERO deductible. If it's an American company, they might well have the option for certificate and documentation in Spanish. Any cover document/certificate that isn't in Spanish will need a certified (by an official Spanish translator) translation.
Personally, when I did it I bought one of those cheap Schengen zone insurance policies (to meet visa rules) it was about 300 euros. I sent the documents to a registered Spanish translator I found on the internet, they translated them and got them certified (probably another 150 euros).
Good luck!