transmission will not shift into third gear

Hello. When you checked the fluid, was it warmed up and in level ground? If not, then you've got to take it around the block and then check it. Aside from that, it kind of sounds like it might be going into "limp mode." What that means is that the computer has detected since sort of issue and is keeping it in second gear. This allows you to drive to a mechanic at low speeds while not burning up the transmission or creating other problems.

The best thing to do would be to get the codes scanned. Even if the engine light isn't on, there could be transmission related codes telling you where to.

If there are any codes, write down the actual code numbers (not just their translations please), and report back here. Also, clear the codes, disconnect the battery for about 5 minutes, and then reconnect it and go for a drive.

Here's a list of codes with their translations. Transmission codes are in the 700s.

https://www.2carpros.com/trouble_codes/obd2.

Let us know what you find and we'll go from there.