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Sounds like the weather-stripper from hell visited your home! I address these issues somewhat in my caulking articles at the website though they refer primarily to bathrooms.
Use a single-edged razor blade at a very low angle in conjunction with heat from a heat gun or hair dryer to separate the caulk from the wood. Heat may help to soften and loosen the caulk making removal easier and with less damage to the underlying wood. Be careful not to overheat the surface… you don't want to strip the finish! Once you get a hunk of caulk started, you might be able to pull an entire strip off… depending on the thickness of the bead. Thicker beads tend to pull off more easily. Sometimes, a pair of pliers can be used to pull a strip of caulk free, though this trick seems to work best with silicone caulk.
Paint remover will also soften most caulks (except silicone), but it can be a messy and dangerous job... plus it is guaranteed to remove the finish on the trim! Liquid Nails markets an adhesive and caulk remover that is somewhat less aggressive than paint remover, but you nevertheless must be cautious and test it on an inconspicuous area of the finish to see if it causes any damage.
If it is silicone caulk, though, you may have a more difficult problem because silicone caulk is not paintable or stainable. Every last bit will have to be removed before you can do any finishing. Even GE Silicone admits that there is no chemical means to remove their product... mechanical scraping is the only way. If you need to repaint, sand the surface first and then apply Kilz primer to get the best adhesion to the finish coat.
If you decide next year that the former owner was right about those windows and you need a better seal, there are a few products on the market that can be used to stop much of the leakage. The first is a flexible, putty-like caulk that is available in rolls. It sticks quite well and works both inside or outside. More importantly, it removes easily from most painted or stained surfaces and can be saved and reused over and over again!
There are also easy-release clear plastic tapes that perform the same function. The adhesive is designed to be removable. Sometimes, though, they don't remove as easily as advertised so I would use them only if aesthetics dictate.
Nail-on permanent weather-strip kits are another option. Designed to work with windows, they provide a year-round solution to window sealing. Though some folks try to use self-adhesive foam strips to perform the same function, they generally gives poor results compared with any of the other options mentioned. The biggest negative is that their permanent adhesive is a bear to get off when the foam inevitably fails!