In my opinion, your plan should be able to work. I think, there can be only one extended partition containing logical volumes, but as you first delete the old one this should be fine. But from which environment do you want to perform this? You have to do it from the system on sda3 (the partition you want to keep), or from a live system. You have to unmount the partitions first, too. And you want to delete the boot partition! You have to mark a new partition as boot volume then. It should be the volume GRUB is installed on.
To later use the new partitions as swap in your fresh installed systems (Ubuntu?) on sda1 and sda2, you can chose them from within the installer to use as swap or later using gparted (right-click on partition -> use as swap(? only know the German title)).
I hope this helps you, otherwise feel free to ask again...
edit: you can perform the repartitioning from within the Ubuntu installer. Just click on "something else..." when it asks whether you want to install it instead of/alongside of your old systems (remember I have the German version, so I don't know the exact inscriptions in English!)
Is this what you want to achieve?
sda1 (primary partition) Linux System 1
sda2 (primary) Linux System 2
sda3 (primary) your old system, "boot" (GRUB installed)
sda4 (extended) <container for logical volumes>
->sda5 (logical) Swap for Linux 1
->sda6 (logical) Swap for Linux 2
But what OS do you have on sda3? Ubuntu? Then you would need another Swap: sda7 (logical)
And you have to specify one boot partition where the BIOS looks for a boot loader (e.g. GRUB). This boot loader then loads an OS you can select in its menu from the partition.
So you can have many OS on your hard disk, but only one active boot loader responsible for all of them.