I had the same problem diagnosed by a riding instructor.   The solution was to go to a bike store and have them replace the stem; it cost around $60.  Well worth it as I felt much safer on the bike.  Hope this helps.

Another cool option is to get an "on the fly" adjustable drop seat post, like a Crank Brothers Joplin R Drop seat post.     You can easily install it on your bike instead of a regular seat post.  You then adjust your seat post length, depending on if you are climbing or descending, without ever getting off your bike.  You just pull on the lever while riding.  Combine that with climbing bars and you'd be set.

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