I'm sorry to hear about the issues that you're experiencing. I haven't had an opportunity to use the exact same printer model that you're using, but I was able to locate the manual online. I'm happy to provide some additional suggestions that will hopefully resolve the problem for you.
It sounds like you've already exhausted your options in terms of paper type settings. This is normally the first thing we recommend checking. I'd suggest sticking with a setting like glossy photo paper or labels after trying the additional recommendations listed below.
According to the manual, the printer model you're using has a function available from the control panel that lets you check the health of the consumable or replaceable supplies within the printer model. This includes the toner cartridges, but may also include other parts such as the transfer belt and fusing unit as well. To access this report you would select "Manage Supplies" from your printer's control panel and then "Print Supplies Status". This will print up a status report that shows the relative health of the various replaceable printer parts. Considering you've tried a new printer model after speaking with HP, this is not likely to be the issue, but it's certainly worth checking just to be sure that all parts are in good working order.
Running a cleaning cycle on the printer is something you should definitely try too. During the printing process paper, toner, and dust particles can accumulate inside the printer and can cause print-quality issues such as toner specks or spatter, smears, streaks, lines, or repeating marks. Printing full color backgrounds at large volumes can exacerbate these types of issues and require the cleaning cycle to be run more frequently. To access the cleaning cycle for this printer model you'll need to select "Device Maintenance" from the control panel and then "Calibration/Cleaning". From this new menu you should choose the "Cleaning Page" option. The cleaning process takes a few minutes, but it should help remove and debris and excess toner that builds up when frequently printing full background colors. The troubleshooting section of the printer manual also mentions running the "Full Calibration" option from this same menu as a possible solution for print quality issues, so while you're in there you may want to do that as well.
You can check out the printer manual that I pulled this information from at the link below:
http://h10032.www1.hp.com/ctg/Manual/c04647409Hopefully one of these troubleshooting techniques will help resolve the issue you're experiencing.