I have is similar issue. I live in Galloway, NJ, but the closest post office is in Egg harbor City (much poorer area). So I have to use the zip code for the poorer area. By just using a zip code search, it's easy to assume I live in the poorer, less appealing area. Many people base there housing searches on what schools there children will have to attend.
Usually a zip code does not span across two counties.
Yes, spanning two counties is not as common. However the zip code/town/school district boundary thing has been discussed here before. Town boundaries don't necessarily follow school district boundaries in California.
Some of the unincorporated or formerly unincorporated areas can also get confusing. El Sobrante is unincorporated and has its own zip code, but parts of it have been annexed by Richmond or Pinole and remain with the same zip code. The post office might still suggest El Sobrante on their zip code finder, but mail should still be delivered if you said Richmond, CA 94803. That probably wouldn't work if you lived in the Richmond District of SF and tried that because of the confusion, just like you might have trouble if you live in the Brentwood neighborhood of Los Angeles (90049) which would get confused with Brentwood, CA in eastern Contra Costa County in the Bay Area.
Yes, this has been a problem for me. I know the post office might not be in the same town as the house itself, so that two mailing addresses will work for the same place sometimes:
123 Shady Lane, post-office-town, CA 98765
123 Shady Lane, geographic-town, CA 98765
But I have everything keyed by zip code. It's very convenient for programming, though it does sometimes piss people off when they see a poorer town listed with their street address, even if that is a valid mailing address for their house. I think it's not a very high % of cases.
Is there any reliable way to go from a street address to a school district? Maybe I should key everything by school district.
I have found an error on the sccoe.org site, where it showed a house in LG school when it was really Union. So it's not perfect, but will be helpful. Certainly in SF Bay Area people care a lot about school district; there can be a $150k difference on the same street depending on district. But in other places it matters less; some places have one school district per county.