I take it you haven't applied for unemployment yet? Dumb you. When a person is laid off through no fault of their own, they need to file for benefits immediately. Unemployment insurance is NOT welfare, is not needs based, and you have no reason to "hold off as long as you can" or as you said, "tried to stay away." This is based on your total misunderstanding of what unemployment insurance is and how you qualify.
Neither will you be declared eligible based on your neediness or your deservingness. You are either eligible or you are not. When you are out of a job and you apply for unemployment, you must be able, available and actively seeking work equivalent to the work you previously held. You must be available for full time work unless you are in "approved training" which they would decide, or unless the work you were laid off from was part time (in some states.) It varies.
The thing for you to do, right now before things go one step further, is that you apply for benefits. You explain to them that you are completing your degree, and that you are available to work, and if you found an equivalent job between now and the time you complete your classes, you would consider whether or not you'd leave the degree or change the classes (IF you found a perfect job) at this time.
In other words, whether you are eligible at this time to draw benefits is sort of contingent on what you say, what you are genuinely willing to do in terms of job seeking right now. But there is NO downside to applying for benefits. The claim will set up now, even if at the present time you are not going to qualify to draw it out because you are not willing to say you'd quit school if you find the perfect job. But why not file? Let them make the decision about your availability.
Why lie? That would be profoundly stupid. Would not make you any more likely to be approved. Be perfectly honest with them. If you are taking night classes, or were taking the same number of classes as while you were working at your last job, I fail to see how your taking the classes now would affect your eligibility. You're continuing to do what you were doing then, which is taking classes while working full time.
As far as why would they find out, if you think you'd be doing something skeevy, you need only to be honest with them about what you are doing. Then a disgruntled friend or ex or somebody who wanted to couldn't report you for fraud. Or you wouldn't be one of the one out of ten or so who was selected for intense scrutiny. Or your income tax records or something tipped them off. They do have lots of resources to find out if you try to lie. Mainly and mostly, the concerned friends you have around you who'd tell on you in a minute if they wanted to are your finest sources of being turned in. They don't understand that it's not welfare, either, and will be very indignant if they think you're committing welfare fraud.
You wouldn't be committing fraud, though, if you tell them you're in shool . File the claim, tell them what you're doing, and let them give you a decision. Don't sit back and try to figure it out by talking to people on the internet before you even file the claim.

