Thanks for your questions. First, can one be consecrated under canon 604 (consecrated virgin living in the world) as well as under c 603 (without making a choice or clear discernment for one or the other)? Once upon a time, in the earliest history of both canons, the answer to that question was yes, but no longer. C 604 outlines a vocation marked by secularity --- a unique and compellingly sacred or eschatological secularity, to be sure, but still, a form of secularity. Consecrated virgins under c 604 are called upon to live this vocation “in the things of the Spirit and the things of the world.” This is a distinctive vocation with its own characteristics and dignity. It is discerned separately from any other vocation and entered into only when one has truly discerned such a vocation. In general, dioceses require that a person come to clarity regarding which vocation they are asking to be consecrated in.
That said, I should also point out that it is conceivable that one makes a mistake in their discernment and after some time (i.e., some years) comes to determine they have a different call. One might also grow into a calling and the eremitical vocation, since it is a second half of life vocation, might be one that one grows into. In such cases one might add profession under c 603 to consecration under c 604, but one would identify as a diocesan hermit and live in that way. If the discernment went the other direction (from hermit to CV), then, after securing one’s bishop’s approval to be consecrated under c 604, one would seek dispensation of one’s vows as a hermit and be consecrated as a CV. It is the case that some have seen that the two vocations can co-exist. I personally do not agree, but given the existence of a handful of such dual vocations now extant, the basic truth in such a case remains: at this point in the canons' history, one must discern which vocation is primary and be consecrated in that specific way.
Moreover, my own impression is that if the two vocations can coexist, it can only occur when one privileges the eremitical over the consecrated virgin calling; that is, they can co-exist only when the eremitical is primary and consecrated virginity adds specific and necessary dimensions to the eremitical life it might not otherwise have. I haven't read or heard anything in discussions of the question, however, that convinces me c 604 has something needed by hermits living under c 603 which their own consecration does not provide. Hermits today recognize the spousal nature of their vocations and often have a profoundly maternal heart which informs and can inspire everything they do. They have these things by virtue of their own personal formation and their consecration by God in eremitical life. The sticking point for me on having such dual vocations is the secularity of c 604 --- significantly eschatological as that may be. Canon 603 call for stricter separation from the world [than other consecrated persons] and that seems to me to conflict with the CV's calling not only to be in the world but not of it as is true for all Christians, but also with the CV's call to act or minister "in the things of the Spirit and the things of the world".
Secondly, you ask if a desire for solitude indicates an eremitical vocation? My answer has to be, perhaps, but not necessarily and not of itself. Your Sisters in the community you left, have a desire for solitude. So do forest rangers, many librarians, and others with quiet and solitary vocations --- all without being called to eremitical life. Unfortunately, for example, so do misanthropes, those with serious clinical depressions, and those with agoraphobia! As I have noted before here, the Unabomber had a strong desire for solitude, but this did not translate into an eremitical call the Church would have recognized or embraced and validated. You get the point, I think. There are many different kinds of solitude and a number of varying reasons for desiring it. Some are healthy and even noble, some are decidedly not healthy and may be downright ignoble. Very few are part of a call to eremitical solitude. When a desire for solitude matures into part of a call to eremitical life, it will also do so beyond the healthy desires for solitude associated with coenobitical life, or normal everyday life and it will do so along with other characteristics which help define it in terms of eremitical life.
I sincerely hope this is helpful! Let me know if it raises more questions or concerns for you.