so I figure I must at least have decent credit.

You should check. Monitor your credit and take advantage of the free annual credit reports (although, they can be a hassle to get).

When I log in to Bank of America's site to do my online banking, it says I'm preselected for their secured credit card.

It is just an ad for one of their other products.

Anybody can get a secured credit card unless they have managed to make themselves so unattractive to banks, that banks refuse to take their money.

I want to apply but I don't want to drop my credit score just to get denied and then have even worse luck getting a different card with someone else.

Check your credit to see what you could get. Some credit monitoring services can give you a pretty good idea.

Don't be too zealous to get credit in the beginning. Credit can take some time to build up and you are right you don't want to apply for something and get rejected. It is a small hit, but it is not very productive and if you can avoid it, that would be best.

On some bank sites, I get ads to apply for things I already own sometimes. It doesn't seem to be the most optimized system sometimes. BofA keeps sending me offers to apply for a card I already have...

By "ad", I mean self-advertising, not third party ads.