Carol… The length of your instrument suggests it is an alto. It isn’t uncommon for players with smaller hands to have a key installed such as your has for the lowest note – an ‘F.’ With all holes covered, without excessive pressure, and a fairly gentle blow into the ‘beak’ at the top, it should play an F. Do make sure that all the sections are pushed all the way in when you put it together.
The grain of the wood and carving suggest it’s a decent instrument. You might look on the front for a makers marking. In your picture I believe I can make out, “Made in Germany” just above the upper joint.
In writing as we are here, a person cannot tell exactly why you’re getting an Aflat or G. There are recorders made for advanced players for just about every note of the scale as their lowest notes. An air leak at the joint from a worn cork or threading can alter the lowest note. There are also instruments made for a variety of tunings for ‘A.’ I don’t have much experience with instruments made for a lower A tuning, but it doesn’t sound like your in that kind of ballpark – at least till you can get a clean lowest note played.
It’s doubtful that the method book you have for C instruments will be of use. Many petite women with small hands find they can play an alto without a key for the lowest note, so don’t jump to any conclusions, but you are harpsichordist and likely can figure your way to playing the instrument on your own. I don’t trust people in places like Guitar Center to touch any instrument I own or to identify yours. Your next best path after your own detective work would be to go online and find a local recorder society or club. After that would be a teacher, even as a ‘walk in’ to a local college woodwind music professor.
Right now I’d say from your posting you have a single-F-key alto of decent quality. There are double-key (F,F#) instruments, but there aren’t that many low F#s around in the literature to warrant disparing of your instrument.
I’m likely covering too much ground and being pedantic, but for what it’s worth that’s all the thinking from this reasonably advanced player. Best of luck. Nice Find! — k —