Even in a higher wattage iron, if you're using one that has the heating element in the handle that heats up a solid tip, you're still going to have a huge issue with thermal recovery when you touch the board.
Ideally, you'd want to upgrade to something with a cartridge style tip, where the heating element is inside the tip itself. They are much more expensive, but I strongly advocate the "buy once, cry once" mentality for something like a soldering station where a lower quality tool can cause more in damages than the money you saved by getting a cheap one. Also why I keep screaming at everyone to buy an oscilloscope before they attempt this potentially damaging repair.
Here's some propaganda from JBC -
. Pay attention to the temperature graphs as they touch a regular through hole solder joint. Tip temperature drops almost 50 degrees until it's taken back off the joint. That's on a 120 watt station with the heating element directly inside of the tip. Now guess at what's happening with a dollar store iron being held against a nearly solid plate of copper.
Now, the JBC stations are probably cost prohibitive for a hobbyist - I pay more for tips than most people pay for irons. Pace, Metcal, and Hakko have some attractive cartridge style options at around $300 USD. The Hakko FX-888D is currently widely considered the Cadillac of affordable irons. It is NOT a cartridge style, but it is very, very capable for only about $100 USD. If that still seems steep, think about how much hassle it would have saved on just this one single use, and then you still have it to use for the rest of your life afterwards.
A preheater would also aid in correcting the issue, but I shudder to suggest that knowing it leads to people putting things in their kitchen oven...