Hi. I have a few things to say.
I assume that the first o-scope image is the gate drive. That's actually pretty good. The remaining ringing is possibly from your o-scope ground but also the 74HC240 package (DIP or SMD) was not designed to be a low inductance package and not designed to operate well at ~7 MHz. (Short version of the story: the limitation is chip internal layout and the wirebonding inside the DIP package. They were not designed for 7 MHz and not designed to be a massive-parallel FET driver.) The frequency indication in the upper right is 6 MHz. I thought you were trying to use 7 MHz? This frequency indication should be very accurate if the trigger is working well.
The drift with time is undoubtedly due to self-heating thermal drift. While you have a nice heatsink on what I assume is the final transistor, there will always be some thermal drift regardless. Do you have heatsink grease or a thermal pad between the back of the transistor and the heatsink?
You should also understand that the HC240 is being asked to deliver (what is for it) a lot of current at high frequency. The Icc/Idd will increase quite a lot just simply because of the frequency involved then to deliver the drive current to the transistor will also create a lot of heat. Most of the heat in this particular internal chip layout will come out of the bottom (because that is where the leadframe, which is what the wirebonds attach to inside, is located) but you have the bottom of the chip pointed up in the air ("dead bug"). (Yes, heat rises but the plastic-to-air thermal interface is never very good at transmitting heat.) You might be able to slightly improve things by trying to put some sort of heat sink or heat spreader on the bottom of the HC240.
There may also be other sources of thermal drift that I am not thinking of right now.
The other thing to note is that the messy o-scope image (the second one) probably has a lot to do with the trigger level you have set. Try moving it lower and experiment to find what gives the clearest result. If that is the output into a 50 Ohm load, it should be a reasonably nice sinewave, once the trigger level is adjusted correctly. Also, check the grounding of your probe. If that is the drain of the transistor, it should
not look like that at all.
If you want to know more about the limitations of the LULU, you might want to see this:
https://www.hfunderground.com/board/index.php/topic,89881.msg288345.html#msg288345(I did see some thermal drift on my circuit as well. I just did not write about it.)