Are you sure that you are using a genuine automatic dishwasher soap, and not just dishwashing soap?
We have just purchased this home. I use cascade dishwashing.
Cascade doesn't usually make suds. Try running the rinse only cycle a few times.
Good idea to run the cycles through without soap, but I would run it through the complete cycle, not just the rinse. This will ensure that the maximum temperature, pressure, and volume of water circulated throughout the machine.
Depending on what type of soap the previous owners used, there may be an accumulated build up of undissolved soap crystals in the machine. I would also add vinegar to the rinse drawer so that this gets passed through all the lines at the end of the wash cycle. See if this helps.
I would run it on high without soap several times. Additionally, each run, I would add 1 cup of vinegar to the top rack in a large bowl or measuring cup with spout. I usually use my 2cup pyrex glass measuring cup. This will allow the vinegar to steam the machine and help clean old residue or soap. Good luck!
Using to much soap do another rinse
Slide out the bottom rack and liberally pour table salt on top of the suds to break down the foam. Run a quick rinse-only cycle and check for remaining suds. If there are still suds in the dishwasher, add more salt and repeat the rinse cycle until there are no more suds. Then toss a tray or so of ice cubes onto the dishwasher and allow them to melt. Push the cancel button to drain the melted ice. Run another quick rinse cycle to remove any suds remaining lines, pump, and drain. Hot water causes suds, freezing cold water does not.
Try using a dishwasher soap that's a solid block or gel pack. They have the correct measure and balance of soap and degreaser and rinse agent, which should solve your problem.