@snacky and
@SunnyDays both hit the nail on the head. Yes, if you or your family is dependent on your working income then you need disability insurance. Period. Disability is WAY more common than people think, becoming disabled is very expensive in medical and associated costs, and the safety net in the US is not timely. I was a super healthy professional when a freak accident left me unable to do any type of work 3 years ago.
SSDI can take years for some people to be approved (and some never get approved). I was fortunate in that my application was approved the first go-round and the process only took about 6 months. There's a bit of money from it but not enough for me to live on without a drastic reduction in lifestyle. I think the main benefit of SSDI is that recipients qualify for Medicare after receiving 2 years of payments. Getting on Medicare has been a huge relief for me.
Having private disability insurance is what has allowed me to keep living independently, save money for my future, and be able to afford the best medical care I can find. I was fortunate to have 2 different LTD policies. One was an individual policy (what it sounds like you're shopping for), the other through my employer. The market for individual LTD insurance is totally different from other type of insurance so there's no utility in going through your usual insurance company State Farm like you mention. Individual LTD is sold through brokers. Important fact: Brokers sell policies but that have NOTHING to do with using the policies so their information about what makes a "good" policy must be considered with a degree of suspicion. Most of the LTD companies are a nightmare but some are better than others so it's important to learn about the companies before you go shopping because brokers only rep certain companies. Brokers will attempt to sell you riders like residual disability that sound good but actually make you less likely to get paid for claims so it's important to know what you're looking for in a policy. So you can save money by not being talked into those. Important policy conditions are things like own-occupation, gender neutral (if you're a woman), no "mental and nervous" exclusion (which are used to deny payments for almost any type of medical condition and not just what you or your doctor might think the term implies), how your pre-disabilty earnings are calculated, ability to increase amount of benefit in the future.
2 great resources to learn about LTD insurance and the various companies are:
whitecoatinvestor.com (I don't have any experience with the brokers on that site and the articles are written mostly by brokers so take with a grain of salt. But lots of useful baseline info and contact info for brokers)
lindanee.wordpress.com (Linda manages LTD claims and has been in the industry forever. She knows how all the different insurance companies operate and has good info about policy provisions, like residual disability riders. She doesn't have anything to do with selling policies.)