It is essential tremor and I have it myself along with myoclonus more often than not.
The best medication for ET is primidone in my opinion. Yes, propranolol is also effective, but if you are already taking a blood pressure and/or heart medication, your doctor will not want to add it along with your current medications. Another class of effective medications are within the benzodiazepines family, in particular chlordiazepoxide, clorazepate, clonazepam, or diazepam.
The reason why I advocate for primidone or those particular benzodiazepines aforementioned is because with these drugs, as you take them, they build up in your system to where if you happened to miss a dose and realize it or not, the metabolites from the original drug has extremely long half-lives (how long it takes to get one-half of the drug out of your system). Yes, these can make you VERY sleepy, sometimes filled with lassitude, and the potential of developing an addiction to the benzodiazepine, but because you have a legitimate reason to take it (because I'm sure you don't want to run out of pills and end up shaking like none other), I'd say the risk is low.
The reason I contend for primidone is because it is similar to the first barbiturate that was discovered: phenobarbital/phenobarbitone. Phenobarbital is used in people who usually have seizures. It's not used as much today because of all these stupid anticonvulsants that don't work and it also is because benzodiazepines phased out barbiturates in the 1960s, particularly chlordiazepoxide and diazepam. If you are more familiar with barbiturates than you are benzodiazepines, I would argue that you ask for primidone versus phenobarbital because, if you live in the United States and/or Canada, phenobarbital is akin to scheduled drugs like benzodiazepines and opioids, meaning that you are taking something that you could potentially abuse. With primidone, your liver metabolizes primidone into phenobarbitone and another metabolite called PEMA that's essentially equal to a barbiturate, but with a weaker action potentially. Also, primidone is not scheduled in either country (I'm not sure about the UK - if you're in the UK, it might be a POM drug), so your doctor can write without having to feel like they are giving you a scheduled drug against something else that you're taking that might make you sleepy. Lastly, primidone is cheaper than phenobarbital and can carefully be watched in your blood and urine tests, should you have to take one.
In my fight with essential tremor and myoclonus, I take both primidone and clonazepam (Klonopin). If I didn't take one without the other, I think that I would still shake and jump like none other. However, I have a high tolerance to both barbiturates and benzodiazepines that my doctor knows that the combination of the both of them may pose a risk, but the benefits they bring keeps me from having to run back and forth to him and the pharmacy to try much more dangerous drugs like anticonvulsants and/or antipsychotics. Don't let your doctor prescribe either class of medicines because they will make it worse - I know because I've ran the gamut of them and I shook MORE on those than I do now with both a precursor to a barbiturate and a benzodiazepine.
Sorry for the long post, but I wanted to explain more in depth than other people about what medications you should ask your doctor to put you on. By the way, there's no real "test" (like drawing blood, biopsy, etc.) to see if you have essential tremor other than the doctor probably asking you to write your name on a piece of paper or draw either a circle or a spiral to see how linear they will be. I hope my information here you can take and research yourself. More should be done to recognize ET as an issue with people, just like CFS/ME should be.