I'm always curious about details on situations like this. Really nerdy stuff like: what's the average housing lot size; what's the average driveway length; what kind of "largest city in the state" (since 20 minutes won't get you from LA to another part of LA and in most places I've seen 20 minutes won't get you far from a city at all which makes it sound like you're in a state where a top-5 city has 100,000 people and becomes rural fast). One of the problems a lot of the US faces is that the cost of infrastructure is shared by so few people: rural roads that have few homes on them, electric lines that have to travel farther to reach customers, sewer systems that require more feet per customer, etc. In a lot of rural places in Europe, people often live clustered in a town center, but rural America tends to spread out a lot more with a lot of distance to cover to connect people. It doesn't really matter how far you are from a city as much as how easy it is to connect lots of households in your area.
As you note, one of the problems that Starlink is facing is that everyone who wants Starlink generally lives in the same areas. Are there a few hundred households that want Starlink within a 7 mile radius of you (150 square miles)? Then there's likely to be capacity constraints, at least until Starlink can launch a lot more satellites.
All three wireless carriers are offering home internet in situations like yours, but with slightly varying offers. T-Mobile has an unlimited offering in areas where it has excess network capacity. T-Mobile also has a "Lite" offering that is available everywhere T-Mobile has coverage, but it has a data cap and costs $150 for the 300GB plan (https://www.t-mobile.com/support/home-internet/t-mobile-home...). AT&T has a home internet offering for $60 with a 350GB cap (https://www.att.com/internet/fixed-wireless/); overages are $10 per 50GB up to a maximum of $200/mo. I don't know if that's $200 total or $200 in overages + $60. At $200, it's certainly expensive, but Starlink isn't cheap. The average home internet user uses around 300GB of data (data from both T-Mobile and Comcast), but that's probably not people on Hacker News. Still, 1TB would be $170 which isn't that much more if you're desperate and doesn't have as high a startup cost (Starlink's being around $700 while AT&T's is up to $150). Verizon also offers home internet service at cheap costs (either LTE https://www.verizon.com/home/lte-home-internet/ or 5G https://www.verizon.com/5g/home depending on where you are), but availability is limited like T-Mobile's unlimited service.
Have you looked into Starlink's RV service? It's more expensive at $135/mo and you're deprioritized behind standard Starlink customers, but that might be a good option. I hate saying this, but once you're in for $110/mo, what's another $25? Well, $300 per year. But if you're desperate, it might be what you need.
Things will likely get better over the coming years. T-Mobile will be covering half of rural households over the next few months with mid-band 5G and they're also embarking on a huge rural coverage expansion with 10,000 new towers coming to improve coverage. Verizon and AT&T are behind on mid-band coverage, but it's coming. Starlink will be launching more satellites which will give them more capacity.
If you don't mind paying an extra $25/mo, Starlink RV might be the best solution. However, people have reported being annoyed that they haven't been able to switch to regular Starlink (and save $25/mo) when capacity becomes available. Definitely check Verizon and AT&T's offering and see if there's something right for you there (and available). It feels like a long-shot, but I'd rather mention them than not even if there's just a 1% chance it'll help. Over the next few years, we're likely to see AT&T, Verizon, and T-Mobile gain a lot of capacity in rural areas as they bring online a lot of spectrum and Starlink will launch more satellites. Right now, since you say you're "desperately looking", maybe grab Starlink RV or even AT&T's service.