2006 Honda Element Wouldn't Shift to 4th

My daughter bought a 2006 Honda Element a little while ago. We don’t have a full history of everything that has happened with this vehicle. It has about 147K miles on it. The other day she told me that the car had trouble shifting into 4th, the RPMs got up to about 4K and when she stopped, the vehicle idled lower than normal (just during that one stop). I had her turn around and head home and it apparently didn’t happen on the way back. The check engine light did not come on at all. Regardless, I checked the codes, looking for anything that might point to the transmission, shifting solenoids, etc. What I got were P0303, P0420, P0304, and P0300. I haven’t heard of these types of codes interfering with shifting before. Could a bad 02 sensor cause that, or a bad shifting solenoid?
The vehicle is somewhat new to us, purchased used, but I think the previous owner had the coil packs replaced recently. Transmission fluid appears to have not ever been changed (it is a little dark but not awful). We had to replace her engine cooling fan and AC compressor cooling fans a few weeks ago. Her engine very briefly overheated at one point while idling and we discovered the fans weren’t working. It was only overheated a few seconds before she turned it off. It has run fine since then (3 or 4 weeks) until today. No obvious signs of a broken head gasket but I don’t have a compression test kit (yet).
Not sure what to do at this point. I don’t want to take it to a transmission place as they are likely to try to sell me a transmission rebuild before doing anything else and I don’t really think it is the transmission since it drove fine later that day. It has been sitting in the driveway for a few days now as my daughter is afraid to drive it. Anybody ever seen this behavior in a Honda before?