Yes, abstract algebraic is a logical next step for you, presuming that you have developed an appropriate level of mathematical maturity from your study of linear algebra and differential equations (e.g. if your textbooks taught theory more than computation).
It is difficult to define abstract algebra in an MSE answer. But one essential point deserves wider emphasis: from an algebraic perspective it's crucial to forget about any internal
structure possessed by the elements of the structure. Such internal
structure is an artefact of the particular construction employed. Such
representational information is not an essential algebraic property. It
matters not whether the elements are represented by sets or not, or by
sequences, matrices, functions, differential or difference operators, etc. Instead, what matters are
how the elements are related to one another under the operations of
the structure. Thus the isomorphism type of a ring depends only upon its
additive and multiplicative structure. Rings with the same addition and
multiplication tables are isomorphic, independent of whatever 'names',
representations or other internal structure the elements might possess.
Andy Magid emphasizes this nicely in his Monthly review of Jacobson's
classic textbook Basic Algebra I. Here is an excerpt:
"This reviewer will not attempt a definition of this essence of
algebraic thought. But perhaps the reader will excuse a little soft
speculation. Algebra seems to be about the holistic properties of
collections of things which, while they have no independent status,
derive their significance from the relations and operations that exist
on the collection as a whole. It makes little difference that the
elements of our ring, say, are matrices, or differential operators, or
formal linear combinations of group elements; in fact it can even be a
positive hindrance to think of them that way. For example, consider the
groups of Galois theory: it is vital, in the end, to think of these
groups as groups of substitutions in the roots of the equation being
examined, but the technical problems which would attend an attempt to
prove the criterion for solvability by radicals while working exclusively
within this particular representation would be enormous. Far better to
ignore the nature of the elements and to think of the group as a thing
in itself. To take another example: the notion that the cosets of a
normal subgroup of a group, while they have intrinsic meaning as subsets
of the original group, are best thought of as unities, as elements of a
new group, the quotient group, is often the pons asinorum of the Basic
Algebra course. Those who cross it successfully usually do learn to
think algebraically. It is probably unfair to claim this thought mode -
ignoring the essence of elements of a structure and focusing on their
relations - as exclusive to Algebra. This is the basis of much modern
abstraction, and not only in mathematics; see for example [Piaget:
Structuralism]. But Algebra does seem to appear whenever structures
dominate a piece of mathematical thought."
-- Andy Magid, Amer. Math. Monthly, Oct. 1986, p.665
-- excerpt from a review of Nathan Jacobson: Basic Algebra I
Thus abstract algebra teaches a sort of structural abstraction, which is ubiquitous in mathematics and its applications. For example, you ask if algebra has applications in physics and chemistry. One of the subjects of algebra is a general study of symmetry by way of group theory. In chemistry this applies to crystals via the study of crystallographic groups, and in physics the Lie symmetry groups of partial differential equations play fundamental roles, e.g, governing conservation laws and separation of variables. And, despite Hardy's speculations to the contrary, even very "pure" fields of algebra like number theory have found important applications (cryptography, coding theory, etc). Also algebraic geometry has many interesting applications (e.g. in robotics and control systems), especially using effective constructive techniques such as Grobner bases. These are only a few of numerous physical applications.
