The Camaldolese invite guests to their monasteries. The Carthusians are extremely strict and do not allow guests unless they are seriously discerning a vocation. In fact, a Carthusian is only allowed to have his family visit for two days a year.
If you feel genuinely called to the Carthusian life, read the book “An Infinity of Little Hours,” it’s an amazing book written by the wife of a man who spent some time at the Charterhouse in England discerning a Carthusian vocation - all the people discussed in the book were discerning a vocation to the life of a cloistered priest-monk, the Carthusians maintain the choir monk/lay brother seperation and the priests spend almost 24 hours a day, six days a week, in their cells doing no required work but chopping wood.
Also, make sure you read as much as you can from the websites of the Carthusian order and of individual Charterhouses.
You may also want to look into the Carmelites, there are a few good-looking Carmelite monasteries that allow for both cenobitic and eremitic lifestyles - one of which in the United States even takes advantage of His Holiness’ motu proprio Summorum Pontificum and prayers the Office and says Mass according to the 1962 rubrics.
Also, consider how much you really know about yourself and being alone. Having to actually confront oneself in the presence of God is terrifying - Mother Mary is the only one who wouldn’t be terrified and that’s only because she never sinned. To quote one of the men from “An Infinity of Little Hours,” “You know you’ve figured it out when you’re lying on the floor of your cell crying.” Don’t make the mistake of thinking life of a hermit is easy. I don’t say this to judge, but simply because I don’t know much about you and don’t know how realistic of an idea you have about hermits.
Which Trappist monastery did you visit, by the way? How traditional were they with the praying of the Hours and how strict were they with silence throughout the day and the Great Silence after compline?
God bless you!