Gloria:
Please, please, please read and listen to MTShooter rather than most of the others. Like him, I have and do live in the shadow of Yellowstone, and visit regularly to photograph wildlife, etc. The vast majority of those trips are solo, and although I don't hike much anymore (I'm a bit over 70), I consider Yellowstone much safer than, say NYC
You will be overwhelmed with photo ops from your car, as there are plentiful pull outs, in addition to the various "touristy places such as geyser basins, etc.". If you choose to, there are many short hikes, and most likely other folks that you can simply ask to tag along with. Again, you are at higher risk of being mugged in the city, than running into a bear. It is simply a matter of being "bear aware" as they say.
Obviously the hours around sunup and sundown are best for viewing wildlife, but don't quit too early. Most wildlife in the park can be seen at virtually all hours of the day. Early September crowds are starting to thin, but as a photographer I have found the following to be true regardless of the season. You will be with other photographers before 9 or 10, then the "others" start to show up. Same goes with after 5pm, the "others" will start to disappear.
You are spot on with regards to the Lamar Valley and Hayden Valley being prime wildlife viewing areas. The Lamar is on the highway to Cooke City, and is thereby traveled regularly (so don't worry about being alone). There are numerous spectacular options for vast Astrophotography opportunities along this route (although the park in general is great for these shots). In September the elk are starting to rut and bugle, and frankly, one of the best places to see that behavior up close and personal is at Mammoth Hot Springs, as a large herd seems to live there year round. If you work at it you can actually get some photos without other people in them.
September in Yellowstone, or Montana in general, is probably different than back east regarding the weather. Historically we will get our first major snow storm the week of the 15th. It typically doesn't last long, but it can be nasty for the short term. Makes for awesome landscape photography. Your clothing therefore needs to address the possibility of temperatures ranging from 15 to 85 (which could happen in the same day
).
All that being said, September is THE most spectacular month to visit Yellowstone and/or Montana.
This will be a trip of a lifetime, so plan accordingly, and expect to be awed. I am every time I visit, as the park is an extraordinary piece of America.
Please don't fear traveling alone, you will soon see that you aren't, and don't plan your trip fearing an encounter with a bear. I have been attempting to get a decent grizzly photo for years, and am still 0 for 1000. Maybe you will have that great opportunity.
Have an awesome trip.