If you want to use your welder on 240vac and you do not have any extra slots look at either purchasing some double single pole breakers for your panel they replace a normal single pole with two breakers and only use one slot. It sounds as if you have a GE panel if you use a THICK breaker probably is a THQL breaker. The thin breaker you purchased was a THQP 20 or 15 depending on the ampacity. Look for a THQP2020 it will use one slot for two 120 volt circuits, you will need two of the 2020 breakers and then you can buy one THQP 220 for your welder on its own circuit. Your panel is probably large enough if you are worried don't dry clothes and weld at same time.