PhD advisor lost funding, need advice

If your advisor can't support you and your research, find another advisor as soon as possible. Your department/grad program should be able to help with this and talking to them should be your next step - it's also possible they can do something to get you funded in the immediate term. In fact, this should have been the advice your advisor gave you instead of suggesting you try to float around for a semester.

You might be able to keep your current advisor as a co-advisor or something, but frankly the way they handled this situation (waiting to tell you until now) is absolutely horrible and unprofessional (and selfish): they would have known long ago that funding was expiring and if they were waiting on some pending grant applications they needed to inform you then so you could line up other support.

I wouldn't be able to trust that person again.

Unless there is a unique situation, if your funding was coming through a grant (private or government), there is a very specific set of rules governing how that money arrives at the university and the "chain of custody". At the vast majority of universities, there is an award contract between the grantor and the grantee, administered by an office of sponsored research (or sponsored programs). That office (and very frequently the school of engineering, the specific department, and the principal investigator, in this case, probably your adviser) knows the exact terms of the contract, including end date, expected work product, equipment budget, students to be funded, etc. Usually, the money goes directly to the university, who then distributes it to the appropriate people/organizations in whatever form necessary (e.g., to a departmental account, as a tuition waiver or from the treasurer to you in the form of a paycheck). There are also university policies governing who in that chain needs to be informed of changes or updates. Your adviser is definitely one of them but you may not be.

If everything is as you described, I can only think of a single scenario in which either your adviser or the department or both did not break any internal policies by telling you so late: funding was pulled because of a major violation of the grant terms (or much less likely, something nefarious). This would be very serious however and even if it partly explained your adviser's desire to "keep this between you", the department and university have a moral obligation to explain it.

All this being said, it is absolutely in your best interest to open conversations with other potential advisers as well as the head of your department, if not the school of engineering. It is likely there are funds to be found somewhere, especially in engineering but it is not likely you can be added to some other grant before next semester, even on what they call a "no-cost extension". If there are no teaching or research assistantships, there may still be one or more department "projects" that you could work on. I wish you luck.