As others say, this GE panel does not support the THQP, their unique type of double-stuff.
Yes, you are past the point of needing a subpanel because you are already consolidating circuits to make space. This is a great example of why 30 spaces is a desperation minimum for a house. Even 42 will leave you with only 7 spaces spare when you unconsolidate the basement circuit and add the welder. Too close to the bone!
That is a fine panel so just install another one right next to it, or wherever else you please. Now you have 64 spaces, and that's a happy number.
You could get clever and make the new sub a 225A, so if you ever upgrade service, you just swaperoo so the new one is the main panel and the original is the sub. Doing the forward thinking to make sure all the details are right is worth the attention of an expert.
Subpanels are well within the range of a well-read home improver. The sub is entirely cold until it is connected to the main panel, where it hooks up like any other load such as the welder you are about to install.
One last thing: we have searched high and low in Code and there seems to be nothing that disallows having as many receptacles as you want on a 30A circuit. So you absolutely can have the dryer circuit power both dryer and welder. Do not extend off the dryer circuit (do a homerun instead) if it is wired with 3-wire cable (hot hot neutral) to a 10-30 receptacle. if it is 4-wire (10/3+ground) cable then you can extend off it.
In fact that is a good safety upgrade to do anyway! The old NEMA 10-30 type receptacle is dangerous, that third pin is neutral not ground, and a neutral wire failure will electrify the chassis of the dryer. Switch the cable to modern 10/3, receptacle and cord to a NEMA 14-30, and sever the neutral-ground tie in the dryer.