Same here, I tell you! I think it's got something to do with the phases of the moon or global warming or something.. Kidding aside, it's quite common for parents with young kids to feel permanently exhausted.
I'm a librarian, so my top tip would be to check if your local library stocks books on sleepy habits - there's usually some great non-fiction titles on sleep hygiene and optimizing rest. Grabbing a good book on the topic might help you uncover some tips to tackle the nodding off during conversations bit; that could be embarrassing!
Also, have you tried napping when your kid naps? I know it's tempting to use that time for chores or leisure, but giving your body some extra shuteye will help. If you don't already have one, perhaps a good neck pillow will offer some comfort and support so you can nod off easily without damaging your posture.
Driving is another story - maybe time to see a doctor? That's a safety issue and should be taken seriously. Don't want to risk dozing off behind the wheel!
Do share any useful tips if you find any, I'm keen to hear 'em!