Seeing the word alcohol on an ingredient list can be alarming, and the instinct to put the product back on the shelf is understandable. But the word covers very different things, and it is worth understanding the difference before deciding, because avoiding every ingredient with alcohol in its name would rule out a great many products unnecessarily.
The alcohol most people are worried about is ethanol, sometimes listed as alcohol denat. This is the intoxicant type, and in the Shafi'i view followed widely in this region, intoxicants are treated as impure. This is the one many choose to avoid, particularly in products that linger on the body or go near the mouth, such as toners, sprays, mouthwash and some perfumes. If your aim is to steer clear of the questionable kind, this is the term to watch for.
Fatty alcohols are a completely different family, despite sharing part of the name. Ingredients like cetyl alcohol, stearyl alcohol and cetearyl alcohol are waxy, non-intoxicating substances used to soften, thicken and stabilise a formula. They do not behave like drinking alcohol at all, and they are generally considered acceptable. Recognising these by name saves you from rejecting a perfectly suitable moisturiser or cleanser on a false alarm.
Between these two clear cases sits genuine scholarly discussion, particularly around synthetic versus fermented sources of ethanol, and around trace amounts used as carriers. Views differ here, and reasonable, knowledgeable people land in different places. This is exactly the kind of question where your own asatizah can guide you for your situation, rather than a blanket rule applied to everyone. Bringing them the specific ingredient and product is far more useful than trying to settle a contested point from a single article.
For everyday shopping, a practical approach helps. Learn to tell ethanol and alcohol denat from the fatty alcohols, treat the former as the one to question and the latter as generally fine, and where a product genuinely matters to you and the source is unclear, ask the brand or your teacher.
For oral care specifically, the simplest path is to avoid the ambiguity altogether. Meraki Halal is a halal certified natural toothpaste made with miswak and black seed oil, and it is alcohol free, so the question never arises in the first place. That is often the quiet appeal of a certified, clearly formulated product: it removes a decision you would otherwise have to make every time you brushed.