In QFT, the exchange of Bosons between certain particles is responsible for the effect of force. My question is how can exchanging bosons lead to a force. Is force defined differently in QFT? (Aka. Should I abandon my classical understanding of force as a contact or non-contact between objects, influencing their motion?)
First define what you mean by "force carrier". It is not a standard term (except possibly in pop physics where it is never clearly defined). If the question is "do bosons have momentum?" then the answer is yes. If question is "do electrons exchange photons when they approach one another in free space?" then the answer is no. If the question is "do electrons interact via the electromagnetic field, and are excitations of that field an example of bosons?" then the answer is yes. If the question is "ok, how is the interaction between electrons described in quantum field theory?" then the answer is something like the following.
The interaction between a pair of charged particles is set out by noting that there are primarily two fields involved: the ones called Dirac field and electromagnetic (EM) field. The interaction can be understood in terms of energy and momentum exchange between each electron and the surrounding EM field, and the propagation of energy and momentum within the EM field. That exchange and propagation is written down mathematically as a complicated integral which cannot be done in one go, but can be expressed as a series of terms. Each term in the series involves the energy and momentum of one or more contributions to the total interaction. This contribution is often called a "virtual particle" (e.g. a "virtual photon") but you should notice (and never forget) the word "virtual" here. Virtual photons are not real photons. They are not even particles. They don't propagate like particles (their quantum amplitude decays exponentially; each one is entirely confined within one interaction), and their energy-momentum involves a combination of energy and momentum that no real photon could ever have. The entities called virtual photons are contributions to a calculation; they help us track energy and momentum and other conserved quantities such as electric charge. They are bosons (this means they have a mathematical property called commutation between certain basic operations in the mathematics).