You really need to do custom lettering for this. If anyone can buy the same font as you and duplicate it just by typing your initials, it's not really a logo, is it? Try sketching what you want and then tracing over it in Illustrator. Or, outline a font that's close and adjust the outlines. Or, use a digital tablet to draw the lettering. Or, hire a lettering artist to work with you.

Good luck!

I agree with Joshua that custom lettering would be better. But here's an example of a "j" with a short descender that can be tucked under the "a" (although I don't think this particular font is suitable for your purpose):
http://new.myfonts.com/fonts/sudtipos/sugar-pie/

@joshua After a few dozen iterations of this logo, I'm starting to agree that I'll have to hire a lettering artist. When I did my first version of the logo long ago I did indeed attempt to draw/trace my own signature, but it didn't go well simply because my handwriting does not a good logo make. It so happens that a friend of mine does lettering and she offered to work with me when I mentioned my difficulties, so I'm hoping I'll finally have a version of my logo I actually like!

@JamesM Thanks! I almost used Sugar Pie, but it was a little too cartoony. Though it suits my illustration style, I'm going for a look that's fun but still professional. I wish I could think of a good example. As Joshua suggested, I'm near the point where I'll have to contract this out.

> In a perfect world I'd have gorgeous handwriting
> and could use my actual signature

Hey, Walt Disney was an illustrator/animator but his famous signature was actually created by one of his artists.

> [Sugar Pie] was a little too cartoony

Just to be clear, I wasn't recommending that you use that font; it was just an example of how a J can be constructed with a short descender.