If you are using an off-set smoker there is nothing wrong with using both charcoal and small, dry splits as a source of fuel. If you have to much smoke taste and not the nasty, sour creosote taste the problem may be that the food is in the smoke for to long a period of time for your tastes.
If you continually have thick white smoke coming out of your smoker you have a problem with the draft and/or the fuel you are using. Make sure your splits are DRY. Dry splits do not burn with thick white smoke for any length of time. I like to use smaller splits and tend to add small amounts of fuel at a time, from the damper side of my firebox. The heat in the firebox gets the new splits burning quickly and moves any initial white smoke through the smoke chamber quickly. You can stack your splits on top of your firebox to let them warm up a bit and dry up any external moisture.
Make sure the meat you start with is dry. Wet meat acts like a sponge to absorb smoke and can come out greasy.
Don't be afraid to pull the meat out of the smoker and place in the oven to finish cooking.
I do any number of things to moderate smoke chamber temps. Small off-sets are a pain in the neck for temperature regulation but a little bit of effort on your part produces some great meals.
Hope this help, sometimes it just takes practice.