I found the blaze spawner and right when I got close blazes began spawning, so I had to run away and block it. Now everything is full of blazes so I can break the wall and go in. I also can't break the spawner because I really need it. I know how I need to put the torches, but how to I actually go in there and do it?
Dealing with Blazes is hard, that's part of the challenge. ;)
It's true that they despawn if you go far away, but of course new ones spawn when you come close again.
The best way to deal with blazes is to get good gear and fight them until you can block the spawner with a lot of light (light level 12 or higher, possible patterns can be seen here and here) or by filling everything around it with blocks.
Especially helpful here are shields and fire resistance potions, because they both prevent all damage from blaze fireballs (but shields of course don't prevent you from stepping into fire).
There are also techniques of slowly approaching a spawner by filling the corridor with blocks and then one by one moving them forwards, but none of them are really failsafe and they take a really long time.