Vernon. Try the recommended dipole to "test" your rig. A dipole for 20 M band is short (about 36 ft long) and easy to make, you will quickly hear whether there is an issue with your K3 hardware or setup. The 20 M band is reliable most daylight hours and shortly after sunset.
You also mentioned that you are still new to HF although you have had your rig for two years. Have you considered trying digital modes such as RTTY and PSK31? Digital modes are able to communicate DX on barefoot and QRP rigs almost as effectively as CW without CW's learning curve. It is a lot easier for me to get through to DX using digital mode than when operating barefoot SSB. Although challenging to learn, no other mode gets DX QSOs through bad conditions as well as CW.
I have logged approximately 10,000 mostly digital QSOs since becoming licensed in 2002. Most QSOs were made on a K2/100 transmitting into a dipole up about 30 feet. You may require a balun on the dipole feedline to keep RF off the coax shield and ferrites on audio cables between your rig and computer to keep RF out of the computer.
There are usually lots of PSK31 signals around 14.070 MHz and 7.035 MHz (40M band). The 20M band is crowded with RTTY signals around 14.080 during every RTTY contest. Try listening for WWV signals and scheduled signals from the ARRL's W1AW.
73 de KB1IKD
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