Nothing could be farther from the truth. Yes, the Democrats are asking to avert the next round of harmful sequester cuts. But they are not doing so as a condition for reopening the government and paying America's bills. After being stuck in the Tea Party mud of the disastrous shutdown, it is the GOP that is looking for a way out - without complete and utter public humiliation. Trying to find that escape hatch, the Republicans have proposed bundling extraneous items.
Sen. Susan Collin's (R-ME) proposal includes two such face-saving measure: delaying the medical device tax under the Affordable Care Act, while locking in the next round of sequester cuts. Democrats had long agreed to the sequester level spending, for now, to keep the government open while budget negotiations continue, but they and the president have also long held steadfast to the position that the Republicans cannot be allowed to extract concessions for that purpose. Keeping the government open and paying our bills is not a concession on the part of the Republicans; it is their job. We do not pay ransom for Congress to do their job. And so if Republicans want the Democrats to give something, like delaying the medical device tax, they need to give something in return - something other than just re-opening the government and stopping the threat of default.
The Democratic side of the argument remains the same: Republicans have two options. They can either (a) pass a clean continuing resolution at current (sequester) funding levels and authorize necessary credit to pay our bills without anything else attached to it - no face saving measure, nothing, or (b) save face by adding something like the delay of the medical device tax while also making a concession to Demcrats, like reduce the harm of the sequester. That Republicans seem to believe doing (a) is "humiliation" for them is their problem, and if it is humiliation for them to do the right thing, the humiliation is entirely self inflicted.
You cannot begin a crisis by demanding ransom just to keep the government open and pay America's bills, and then complain that you have to be given something so that you can get out of it and not look humiliated. You cannot go down that road and when you get blowback from the American people, pout your lips and complain that the people who always wanted to keep the government open and pay our bills aren't giving you candy for your agreement in doing so. The reason for GOP's ritual humiliation is not that Democrats refuse to pay ransom, it is that Republicans demanded the ransom in the first place.