You mix the notion of a molecule that stores information that can be copied a.k.a. replicated, like RNA, with the notion of a self-replicating system, like a living cell.

Like another poster has said, evolution cannot exist, unless you have a self-replicating system. The meaning of evolution is that the replication process is imperfect, so the replicas are not completely identical to their parent system, i.e. they evolve. It is meaningless to talk about evolution unless you talk about a self-replicating system.

The existence of information-storing molecules, like RNA, is not necessary for a self-replicating chemical system. In fact it is pretty much certain that information-storing molecules did not exist in the first living entities.

In a self-replicating chemical system without nucleic acids there must exist a closed cycle of chemical reactions, each producing reactants needed by reactions located later in the cycle and where the net result is the conversion of the simple molecules from the environment, like carbon dioxide, dihydrogen, ammonia and sulfide or sulfite into complex organic substances, including peptides and organic acids, which are either catalysts or reactants or products for various reactions along the closed cycle. Such a closed cycle of reactions is likely to have appeared in pores of suitable minerals, like metallic sulfides, in hydrothermal vents or volcanic vents that supplied the input gases.

The difference between the first self-replicating chemical systems and the living beings with nucleic acids is the same as between hardwired control automata and microprogrammed control automata (where the control information is stored in a microprogram memory). The use of a memory (i.e. nucleic acids) has greatly accelerated the evolution of living beings, because previously any changes in the components of a self-replicating chemical system would have been very likely to result in a system that could no longer replicate, so it would die without descendants.

There is no doubt that DNA is a later invention and that initially there was only RNA. RNA has 2 main functions, it can be copied into another RNA molecule or it serves as a template for the synthesis of a protein molecule.

There is no doubt that the protein synthesis function has appeared later than the self-replication function. The reason is that if there would have ever existed an RNA molecule without self-replication that could be used for protein synthesis, it would have immediately disappeared without any descendants.

So the first nucleic acids were RNA molecules able to be replicated, but without any useful function. In other words, they were RNA viruses that were parasites of some self-replicating chemical systems which had metabolism, but which did not use nucleic acids.

The host systems must have been using ATP for a long time before the apparition of replicable RNA, and RNA must have appeared due to side reactions that caused losses of ATP by polymerization (ATP and related nucleotides are just monomers of RNA, i.e. its constituent blocks). ATP must have been used as a dehydrating agent to create peptide bonds long before the use of nucleic acids as templates for protein synthesis.

Even in the living beings of today there are many peptides that are created without being synthesized in ribosomes with RNA templates, but the mechanism of their synthesis is much less known and understood than the synthesis of proteins with RNA templates. Before nucleic acids, the catalysts of the biological reactions must have been such peptides, which used a much smaller set of amino acids than the proteins of today, perhaps only about 6, but certainly no more than 10.