I would never write a letter at all. If I got such a letter, I would think it was too weird.
I would advertise in a local paper about wanting to buy and my requirements, leaving out the price considerations, then go immediately to check it out if someone contacts me If I found a place that seemed within negotiating distance, I would telling the owner how much I like what they have done and I value it too.
At least, I got my second house that way.
Another person got his house by stopping when he saw the owner in the yard to tell them he liked their place so much and, if they decided to sell, would they please contact him first? It was a year later but they did work themselves into deciding to sell and were happy to avoid the hassles of marketing by calling him and negotiating a sale that did well by both parties.
Glad that worked for you! Advertising is one way to attract Sellers. Then, the Seller can choose to contact or not.
While some folks prefer not to use Realtors, Agents, or Real Estate Attorneys, this is one situation where it would be a much greater benefit to the Buyer. Why? Because the above post nails one point (I bolded it). A letter of Intent to Buy is not usually seen as a scam. Why? The Owner can check out the Realtor, Agent, or Real Estate Attorney. Of course, the Buyer has far more protection utilizing a Real Estate Contract, approved by the State involved. I keep reading all sorts of stories - Buyer regrets. I would never buy a property or home without utilizing a Realtor, Agent, or Real Estate Attorney. With ten years of experience in real estate, not worth experiencing all the nightmares. Of course, there are good happy stories, but less of them.
I also would never give a Seller any personal information about myself.
Sure, one can show up at homes to stop and chat with the Owner of a home he or she was interested in buying. However, that is a risk, as well. Unless there was some sort of rapport established, the Owner could also call the Police and report the interested party. If a complete stranger showed up at our property, I would politely request that person depart immediately. If the person was just driving by and stopped, I would simply say thanks and never contact that person to purchase my property, even if I wanted to sell later. A neighbor or someone I knew personally, yes. We do have a property that isn't in a neighborhood, as it is rural. No one is welcome on our property without being invited by us first. The only exception is a neighbor. We had two different guys drive up our driveway, get out of their vehicles, and talk to us. The first one was selling something and I never buy from any one who trespasses on my property! The other guy wanted to introduce himself and gave us his business card. Too bad, the guy doesn't know I would never hire someone for a service he or she just showed up, trespassed on our property for the sole purpose of selling a service.
I may be a Realtor, but I do not believe in being intrusive, cold-calling, trespassing, or harassing folks in any way.