No. Modern SSDs are quite sophisticated in how they handle wear leveling and are, for the most part, black boxes.

SSDs maintain a mapping of logical blocks (what your OS sees) to physical blocks (where the data is physically stored on the flash chips). For instance, when your computer writes to the logical block address 100, the SSD might map that to a physical block address of 200 (this is a very simplified). If you overwrite logical block address 100 again, the SSD might write to physical block address 300 and remap it, while not touching the data at physical block address 200. This let's you avoid wearing out a particular part of the flash memory and instead spread the load out. It also means that someone could potentially rip the flash chips off the SSD, read them directly, and see data you thought was overwritten.

You can't just overwrite the entire SSD either because most SSDs overprovision, e.g. physically have more storage than they report. This is for wear leveling and increased life span of the SSD. If you overwrite the entire SSD, there may be physical flash that was not being overwritten. You can try overwriting the drive multiple times, but because SSDs are black boxes, you can't be 100% sure how it handles wear leveling and that all the data was actually overwritten.

No, "overwritten" data doesn't actually get erased right away due to wear levelling. As SSDs get esoterically smart with how they prevent unnecessary erase operations, there's no way to be sure without secure erase.