I didn't read all the replies, so I apologize if others have already made similar suggestions.
You might try modding the old bike. Might be able to flip up the stem (or even replace it with a shorter one) to get a more upright and comfortable position. Might also be able to squeeze slightly wider tires into the frame. This might give more comfort, but might not address the "twitchiness."
If you do want a new bike, the market for stable, comfortable distance bikes has blown up in recent years, so you have many options. Look at bikes marketed in the "endurance" and "gravel" categories. I'm 56 now, and I'll never buy another road racing bike again.