I am so sorry about your mom.
Your sister as the executrix needs to hire/consult an estate & probate attorney ASAP, and do things right. They will instruct her for the laws in your state and may be able to help get you both where you want to be in so far as owning the home together in your own names.
Why are you so sure you and sibling won't qualify for a mortgage? If you are both employed, have halfway decent credit or have assets to pay, then you need to at least try to get a mortgage before assuming you won't. But if you can't get a loan to pay off the current mortgage, then the house needs to be sold, or you will need to find assets to pay it off elsewhere.
Just a note about being a executor: Not doing her appointed job - acting as the legally responsible entity for settling your mother's estate - means your sister is failing her legal duty, and she could be found in contempt of court. She can be fined, wages garnished to pay back the estate, and in extreme cases, face jail time. The last is not likely, but she needs to understand the level of what she has promised and do it legally. She must pay off all debts - like the mortgage - as long as there are assets there to do so as part of her responsibilities. (I know this sounds harsh, and I am not intending it to sound cruel, but it is a very real serious concern that you both should be aware of – I absolutely want things to work out for you both)
As far as the living in the house with it being in your mother's name/mortgage, no you can't keep up with that indefinitely. While it might be possible to drag things out for a year or two if all heirs conspire to do so, ultimately you need things paid/settled and transferred to the heirs and close out everything and file the estate taxes. The longer the estate drags on, the worse and more complicated it's going to be. Keeping up with accounts/taxes, forms and policies and property (and insurance!) isn't fun even when it's your own stuff.
Something else you haven't thought of: you can't count on insurance on a house owned by a deceased person, so that right there is a HUGE deal. The current insurance policy is in your mother's name, so the existing policy is no longer valid. It MUST be rewritten for the estate/current occupants. You could be in for a very nasty surprise if you had serious home damage/someone sued you and the executor did not take care of this. This is why it's vital for the executor to do their job: if the house got damaged 6 months from now and your sister did not actually get the insurance updated, then you would be on the hook for the costs of repairs. And if the mortgage company discovered that the house wasn't legally insured, or they discovered that the actual owner died years ago - the could swoop in and sue you for fraud, and/or foreclose on you. There are tons of nasty things that could happen if you don't get things legally squared away.
I get that you both are in mourning. You just lost your mother. It would be fine for your sister to just keep up with the basic bills for a month or two until she can work up to dealing with hiring an estate lawyer. But she needs to do the job, and if she can't or won't be able to deal with all of this, then you should help her find someone that can act as a guide so you can get this settled as painlessly as possible.
ETA: you're out of debt according to your post from earlier this year, have a not shabby credit score (above 700) and some emergency funds. The only thing that may impact a good price on a mortgage is your lower than average income (at least you do report/pay taxes) and potentially your sister's credit score/debts. If your sister has decent credit and at least ~30k income (roughly what you stated yours is) then you do stand a VERY good chance of getting a mortgage enough to pay off the $150k your mom took out on the property. If it was a recent refinance, then the appraisal is still good, so there's some decent equity there.