I think you should change them.

You can just use radial's and bend the longer + lead over.

I know the axial's are harder to find locally and usually more expensive.

This issue might be a noise issue because of the degraded caps in the PSU, pushing more noise into the main board,

and because those 2 caps you you have not replaced are allowing the noise because they are degraded as well.

Also make sure to wipe the contacts of the RAM its self, sometimes skin oil + dust can make a poor connection.

But at-least your computer is mostly working… you should be happy for that

Could you post a picture of your main-board when you get a chance?