I'm an owner of a 145.022-76 which, according to ilovemyspeedmaster, was manufactured in the summer of 1977. This watch was my first vintage purchase and I'm assuming this is your first one as well. I think I did things the right way and got a great watch, so I think I can offer some insight into this based on my experience.
1. Make sure to do plenty of research on condition and originality first. I'd recommend reading speedmaster101. There's lots of good information on the site about how to determine if a particular watch is original or not, with regards to the bezel, the hands, the fonts, making sure you see the etched Omega logo on the inside of the crystal, etc. Regardless of how much you trust any dealer, you should always do your own research and make sure you feel comfortable with a purchase, so you have peace of mind. In my experience, having all this knowledge also enhances the experience of ownership, since you know more about your watch and what makes it great.
2. I would personally recommend prioritizing condition above both the year and whether box and papers are included with a watch. If the watch being manufactured in 1977 and having its box and papers is incredibly important to you, there's nothing wrong with that, but ultimately, your enjoyment in wearing the watch is going to be a result of its condition, not its year or the box and papers. Box and papers can often be faked. If you're debating whether to spend an extra dollar on condition or box and papers, I would go with condition every time. Watches in better condition will also hold their value for longer, which is an important thing to consider. For example, my watch has a faded ghost bezel, and I absolutely love that since it's rare and part of what makes my watch a great piece. Things like this make a watch special and unique, so you should prioritize those over everything else.
3. I would consider reaching out to prominent and trustworthy dealers on the site and elsewhere to have them source a watch for you, since your looking for something pretty specific. You can look up seller reviews on the site, and you can reach out to someone like Eric Wind, who runs Wind Vintage. I haven't dealt with him personally, but he seems like a trustworthy guy. Dealing with a dealer like this will be costly, but since you're looking for something very specific, it might be worthwhile since you'll probably have to wait a long time for a 1977 Speedy with box and papers to just show up on a particular site.
4. You should look up the watch you're looking for on watchrecon.com. Put the reference into the search tab and you should get a compilation of all the watches currently for sale across different forums and sites. You can also see the prices of watches that have previously sold.
Feel free to message or reply to this with any more questions. I'm happy to share my own experience of hunting for a 145.022 and I may be open to selling my own piece as well.
Here's my watch: