It's a tricky situation. But I strongly advise you not to lie. it is not unknown for there to be two interviewers in the interview room, and the second one says not a word but is observing your body language. Lying is risky under those circumstances, even if a "family illness" etc is hard for prospective new employers to verify.
If you can get some good job references from your previous two employers, you could take those to your next job interviews and tell them the absolute truth, as you have framed it in your question. All of a sudden you will come across as a person who not only is competent in the workplace, but also has sufficient self-respect to maintain your high personal standards of behaviour in the workplace.
There are companies who don't, as policy, issue written references or testimonials. But there will always be someone at your two previous "good" workplaces who will accept a phone call from a prospective new employer. If there isn't a formal HR division at your previous workplaces, then choose a "first-choice" and second choice name, of people you worked under or with, just so long as they are people of some significance in the business (eg.don't name the car-park attendant as a referee!).
Given your circumstances, you really want to make sure a prospective employer gets a good testimonial whether written or verbal. Otherwise they may not bother any more with you and go to the next person on their list of new applicants for the job.
Here's another way to get around it. By any chance have you done charitable work? you see what I am getting at. It isn't job experience, I imagine, but it could be an indicator of how you can work with others co-operatively and supportively. "Big name" charities would be best, but any charity respected by the community-at-large will suffice. With this suggestion you are doing your best to focus your prospective employer's attention on your overall capability to get along with others, as well as, of course their primary concerns that you are work-ready.
And remember, you are one year older, and if my guess is right as to your age, one more year will readily be seen as a significant and perfectly normal step into greater maturity. It would be perfectly logical to want to settle down in to a career path about now.
As a graduate I am sure you can follow the several lines of argument I give here. You will have the intelligence to weave the threads that will work for you, into a cohesive whole which is both perfectly accurate and which is purposely designed by you to maximise the good and competent parts of you.
Self-respect got you into an unhappy state of unemployment and no fault should be ascribed to you. But you still have to come up with every honest "weapon" at your disposal. So here are another couple of last-minute thoughts just to make I have given you "everything":
What does your priest think of you? Based on what evidence he has seen from you?
I have made an assumption that you are a woman? can you apply for jobs with a woman GM or owner or majority shareholder? In this case it is certainly not sexism, it is seeking out the half of the human race who can understand your situation like no other.
I don't like wishing people good luck. I wish you "good skill".