Get the amp 'cutting out part' fixed first. Then focus on the cab replacement.
Go through all the tubes first. One at a time. Use Dioxit contact cleaner or similar. No WD40 or greasy spray as that shit will actually collect more dust and dirt. Spry the tube pins and then work the tube in and out of the socket to clean the sockets and pins. Preamp tubes and power tubes. Then you can fire the amp up unplug the guitar. Tap on each tube lightly with the eraser side of a pencil (for example). If you can hear the tapping through the speakers then you MAY have a microphonic tube. This is not a 100% guaranteed test but it is free and you can do it yourself. Google this method for more into. I actually created a video on this but I made the mistake of uploading it to Photofuckit, and now it is gone. You can also do the Dioxit trick for just about every single jack in the signal chain including the speaker cable. Spray on the jack tip and then work it in and out of the socket(s). I do this at least twice a year when I go through my amps and do this stuff above.
Again - doesn't cost anything outside of a can of contact cleaner which should last a long time. IF you need new tubes then order up a few preamp tubes and swap a known good tube through V1, V2, V3, PI, etc until you find the bad one. Google this too - there is tons of info out there. For power tubes, make sure you order a matched quad that fits within the parameters of your amp (roughly). A bias probe/multimeter is also good to have if you don't have one but that is a different deal. Always have extra tubes on hand for testing and swapping. If you already know all of this - then I apologize but I'm going to copy this post as it comes up frequently. PS: I'm not an amp tech and I don't pretend to be the expert.
Lastly, look at a different cab as mentioned. All good rec's in this thread. I have a Mesa Trad Slant 4x12 with Mesa branded V30s and it is so awesome. I also had an Avatar cab at one point and those are really good cabs for the money and you can order their Hellatone brand speakers (I think still) which are just Celestion speakers that are pre broken in and actually cheaper.
Good luck and nothing wrong with a DSL imo.