Given the number of people intending on raising fire-types from birth, fireproof furniture coverings are actually a lot less expensive than you might think.
On a serious note, with all fire-type hatchlings, it’s imperative that you rely on their fire-type parent or other experienced fire user to help you. The problem is that while we can train our pokémon to understand and use their element as best as we can, we’re still only human with (presumably) no actual, inherent ability to control flame. It would be like if a slowpoke tried to teach another slowpoke how to talk; we just don’t have that nearly instinctive understanding of the elements or how to control them.
However, pokémon do. While it’s true (and quite obvious, as you’ve noticed) that not all of them are born with complete control over their elements, they eventually gain that understanding as they grow and unlock their full abilities and powers. Thus, a pokémon who’s managed to reach their final evolution (or who is at least several years older than a hatchling) would logically understand their element better than any human ever can. Or in other words, you and your charmander might not know how to control fire, but your charizard certainly does.
Explain the situation to your charizard (yes, it will understand) and set up a specific time to have it train your charmander every day. Given the immediacy of the situation and the danger fire presents, I would actually recommend setting up a training session that goes for as long as your charmander can handle—that is, at least and absolutely not less than one hour a day. As your charmander gains better mastery over fire, you can reduce this training time down to the standard hour-per-day most pokémon train for, if either your charmander or your charizard wish. Also, ensure these sessions happen in an open area with very little brush or other plant life that could catch on fire. Empty parking lots might be a good idea if you can’t reserve a battlefield at your local pokémon center. Bring with you fireproof targets (cinderblocks, bricks, or rocks will do, and keep in mind you can request such targets if you’re working at a pokémon center), and once the training grounds are set up, step back and let your charizard take it from there.
Best of luck, and congratulations on the new addition to your family!