Let’s start with a basic question, did you check the transmission fluid level? If it is low, one of the symptoms is just what you describe. To find out, run the Toyo for about 20 minutes, park it with the handbrake set. Then, with your foot on the brake, the only way the transmission will work correctly, push the transmission through each gear setting for a moment of two to get things lubricated correctly. Finally, pull out the tranny dipstick, wipe it and reinsert; then take a reading. It should normally be between the low and full marks. If it is below the low mark, you have found the reason for the poor shifting.
To fix it, get a couple of quarts of the proper transmission fluid (if the manual reads Dexron or another similar brand, don’t try a substitute, use what is required) and a funnel. Then, pour, a bit at a time, the fluid back into the transmission, checking often to make sure you are not overfilling the transmission, which is just as bad as underfilling.
Another potential problem that can be shown by draining the transmission and dropping the pan to get out any remaining fluid is a plugged filter and screen. If the filter is plugged up to the point where the fluid flow is cut way down, then it is possible that your transmission will act in the manner you describe.
If you find this is the case, replace the transmission filter and screen and replace the transmission cover, using the proper sealer before you insert the screws. Then refill the transmission with a fresh load of fluid and you should be good to go.
One other suggestion, but this is getting into the area where you will need a tech to tear down your transmission, unless you have the facilities to drop the transaxle, is the valve body. If one or more check valves is either lazy or stuck, then your Toyo will perform as you describe. Let me know if this is the case and I can direct you from there.