Could be one or more of a lot of things, but this is what I would check in order:
1) Depending on the age, even if the Power tubes are not bad, they will be eventually. Have you noticed the sound not seeming as loud, or you have to really crank it to get any volume? A Fender Twin should be pretty loud on about 3-4, if you have noticed having to go up to 6-7, I suggest your tubes were going. Power tube's generally die slowly. You will be stunned, amazed, stupified, elated (and any number of additional descriptions) when you put in a new set of tubes and fire it up. You will have one of those angelic chorus epiphanies. Unfortunately, you will likely have to bias the amp, so unless you are comfortable with this process, you should probably find you a tech.
Since it is effecting both channels, it is likely either the power tubes or...
2) The Phase Inverter tube, which depending on which version of the amp you have is likely a 12AT7 (or 12AX7). This is probably V4, again depending on the circuit. You should buy some spare tubes anyway, 12AT7's and 12AX7's are cheap. Here the
NOS Philips 12AT7 are available for a great price...I would definitely buy one or two (in fact I have!). If it is a 12AX7, just buy a few of those, I like JJ's or Ei's (if you can find them) for Phase Inverters, but any brand will work. If you are buying for the pre-amp tubes, V1, V2, etc, I would go with the Tung-Sols or Mullards if you don't want to pay the extra for NOS. Just buy some and rotate them around. You will be surprised how different the tubes in V1/V2 can effect the "character" of the amp. Point is, tubes go bad, you have to replace them.
3) If it is not the tubes, then next most likely from the sounds of what you are describing, would be something in the power section. I would check the B+ voltages, to see if you are getting the appropriate range of DC Volts to the plates of the tubes. Most likely problem here is the diodes that rectify the AC from the transformer into DC for the tubes. Either these or the Filter Caps are shot. Depending on the age of the amp, the filter caps will need to be replaced anyway. Again, unless you know your way around, I would probably take this to a tech.
Start with the power tubes and phase inverter though. 9 times out of 10, that will fix what ails you when you are getting what you are describing.
On the tubes:
I would suggest you not be constrained to sticking with the "Fender" branded or Groove Tubes. I know you want to keep the "Fender" look, etc., but people don't "hear" the Fender labels on they tubes, but they can definitely "hear" the tubes.
Any 6L6GC is compatible, most 5881's are. I think the Groove Tubes are either the Sovtek or Shuguang/Sino (Chinese) brands if I recall...which while decent, most people would agree that neither of them are as warm and full as some of the other brands. Some call them flat. They don't work for me...in fact, I tend to recommend to my friends that buy "new" Fender amps, to pull the Fender tubes and store them for when/if they sell the amp. I really, really like the Tung-Sol 6L6GC reissues. Their 5881's are sweet too in tweed style Fenders where you want a bluesier early breakup.
Anyways, food for thought.
Craig