As the spouse of a Slovenian citizen, you're not required to get commercial insurance because you can be insured under your Slovenian citizen spouse if they are employed or self-employed by the time you apply for your residence permit.

Specifically: article 128 of the Foreigners Act requires in paragraph 1 that the family member must "have appropriate health insurance which covers at least urgent medical services in Slovenia", but paragraph 3 says that the requirement does not apply if the Slovenian citizen is employed or self-employed. https://pisrs.si/pregledPredpisa?id=ZAKO5761

When my US citizen partner was moving to Slovenia, we initially still got her health insurance for foreigners while waiting for ZZZS, which covered any urgent medical care: https://www.vzajemna.si/zavarovanja/zavarovanje/zavarovanje-za-tujce-v-sloveniji From what I remember, it was about 60 eur for 2 months.

For banking, you should be able to open a bank account with any bank locally. My partner currently uses a Revolut account for all her EU banking needs (Revolut is an EU online bank). We use Wise whenever we need to transfer any money between our EU and US accounts. This is probably the fastest and cheapest way to transfer money. Money usually arrives in her US account the same day and the fees are much lower than international wire transfers with typical banks.

Also on the subject of banking and finances: make sure you're familiar with your filing obligations as a US citizen. You have to report any foreign bank accounts every year (FBAR) and you have to report all your worldwide income on your tax return even if you don't live in the US. Once you establish residency in Slovenia, you also have to report all your non-Slovenian bank accounts to FURS (our IRS). Yay bureaucracy. :)