Steven W. answered 07/21/19
Physics Ph.D., *professional*, easygoing, 9000+ hours tutoring physics
The error is that the two forces you describe are not a Newton's third law action-reaction pair. Say that the first spring you mention is compressed so that is exerts a 10 N force and then you place it next to an uncompressed second spring. The first spring will exert a force of 10 N on the second spring. What Newton's third law says is that, solely because of that 10 N force, the second spring -- even uncompressed -- will exert a 10 N force back on the first spring.
The result of this is that, even if the second spring is uncompressed, and you let loose the first spring next to it, the first spring will also shoot away in the opposite direction from the second spring (absent any other forces), due to the reaction force alone.
The same thing would be true if the second spring were compressed so that it exerted a 7 N force, and was placed next to an uncompressed first spring. The first spring would still exert a 7 N force back on the second spring BECAUSE the second spring exerted a force on the first.
Now, if both springs are compressed, then each one will receive both the reaction force from the other spring AND the force the other spring exerts due to its compression. So the net force on each will be 17 N (in opposite directions, as Newton's third law prescribes).
The way I usually paraphrase Newton's third law is: "If System 1 exerts a force on System 2, then System 2 exerts an equal force, in the opposite direction, on System 1." Note that conditional phrase at the start. Newton's third law ties a reaction force to the presence of an instigation force: IF System 1 exerts a force on System 2, then there is a reaction force. The reaction force is always in response to what System 1 exerts.
I hope that helps! If you have any follow-up questions, please do not hesitate to ask.