I've built 6 mods over the past few months, sealed 510 connectors from MV in each. I never take the connector apart. My usual method employs a pair of slip-joint pliers (opened to widest) to hold the connector, a rubber band wrapped around the handle of the pliers, and a soldering iron around 25W - with very sharp pointed tip. It will help you greatly to have your wire pre-tinned before you try to solder to the connector, leave plenty of solder on the wire when you tin it. Pre-tin the pin on the 510 connector, shouldn't take more than a couple of seconds to transfer sufficient heat since the pin is so small.
With the 510 connector held securely by the pliers, press the end of the wire to the connector with one hand, touch the soldering iron to the junction of the wire/pin with the other hand. Remove the soldering iron, hold the wire to connector being as still as you can and you are done.
If you weren't steady enough holding the wire to the connector the solder joint will look dim, not shiny and not smooth, which would indicate a potential cold solder joint. If this happens just touch the soldering iron to the joint again, being sure to press the wire to the connector as you do so, remove the iron and hold everything completely still. A good solder joint will be visibly shiny and smooth. Be sure not to create a solder bridge from the center pin to the shell of connector.
With proper technique no solder connection should ever require touching the soldering iron to the connection for more than a second or two. The only exception would be when soldering something large that will dissipate a large volume of heat.
I worked as a FCC licensed technician for 10-plus years so I have extensive soldering experience on component-level repair of all types of equipment, but that was more than twenty ago, and it is an easy skill to master. I do have to wear my reading glasses so I can see everything clearly though.
And, by the way, if you don't have a sharp pointed soldering tip you can make one very easily. Take an older tip you have, and with a file shape it to the profile you want. This will ruin the tinning that was applied to the tip at the factory, but you can re-tin the tip by heating the tip on the iron, applying solder to cover thoroughly, wiping excess solder with a damp sponge, and unplugging the iron and letting it cool completely. Repeat this process two or three times and the tip is tinned.