Modern O- or X-ring chains come with grease inside the rollers. The O- or X-ring is intended to keep that grease in. If you're oiling or greasing the chain, all you do is keep the O- or X-rings lubricated, and prevent rust on the outside of the chain. But whatever you put on the chain does not (re-)lubricate the chain itself.
If your chain isn't moving anymore in spots, it means the grease inside the rollers has gone, and they have now seized. This eventually happens to every chain, but the better you maintain it, the more you can prolong the inevitable. But when it happens, the chain is end-of-life and should be replaced.
In fact, if you look carefully, there's a really high chance that some, if not most, of your O- or X-rings are missing now. Also, without grease inside, the internal parts of the rollers wear away which leads to apparent "stretching" of the chain. There is a test in the owners/service manual where you measure a certain number of links, and if their combined length exceeds a certain value, the chain should be written off.