Generally, how hands-on ME is compared to EE depends on the curriculum in the particular university you are in. Though, generally, ME might be more hands-on than EE. In EE, core hands-on courses usually include a digital circuits lab and an analog circuits lab, and perhaps a requirement for everyone to take an advanced lab course from a list of topics in their senior year, and maybe an additional hands-on capstone project. EE programs usually also have at least a programming course as a requirement.
If you study ME, you will be able to crossover and take some of the EE courses, but as you proceed to junior and senior years, this might become harder for certain courses since the courses tend to become more specialized and have more advanced prerequisites, such as some hands-on courses that require prior experience with certain lab skills. Though, as I will explain in the next paragraph, there is some overlap between these two fields, especially in the more analytical areas.
In terms of utilizing your analytical and problem solving skills, both EE and ME could be good choices for you as long as you choose your coursework properly; though, in general, the core courses in EE programs in many universities have a slightly more theoretical flavor. For instance, the areas of control theory, communication (including information theory), and signal processing, which are typically considered as core areas in EE, are some of the most analytical areas in engineering that require a solid mathematical foundation. Control theory, and even signal processing, are widely used in ME as well, and depending on the university you are in, they may be considered as core topics both in EE and ME (as well as aerospace engineering). However, both EE and ME also include topics that do not really overlap much with each other, and these topics could be theoretical, hand-on, or both. For instance, EE includes topics in electro-physics such as solid state devices, photonics, optics, electromagnetics, etc., which could be both analytical and experimental, depending on what type of advanced coursework you pick. While some of these topics are related to other engineering fields such as materials engineering and even mechanical engineering, many of their aspects/applications often tend to be considerably more specific to EE than any other engineering field.
Considering that you have good analytical skills, I should also add that with the advent of big data analytics, fields such as electrical engineering and computer science (CS) that build a strong mathematical foundation for students at the undergraduate level are becoming very popular, particularly among those students who find themselves more interested in engineering than in pure math or statistics. A double major in EE (or maybe ME if you like ME better) and CS (and maybe a minor in math or statistics) can build a very strong background at the undergraduate level in terms of utilizing your analytical and problem-solving skills.
As for your interest in robotics, energy, and environmental sustainability, both EE and ME are closely related to these areas; so, either way, you'll be able to get into these areas. The only difference would be the type of problem you will be solving in any of these areas, which would depend on whether you have an EE or an ME background.