One interesting solution I have played with for paradoxes, is the "quantum universes" approach.
An event that would remove the causal forces that led to the event, (Such as inventing a time machine to save a lover from a tragic death-- Once you save them from the tragic death, the motivation to create the time machine goes away, and it never gets built, meaning you cant go back in time to prevent the tragedy-- Boom-- paradox) can be seen as both having happened, and having not happened. You wont know which universe this is until you measure. Very much like a quantum particle's state. The two universes are in superposition until observed. You can expand this with Many Worlds type thinking, and say that both universes definitely exist-- the question is just which one you are currently observing. The universe that does invent time travel, does so, and saves the loved one from the universe that does not invent time travel. The major issue comes from trying to be rational with the preconceived notion that there is only one universe, and not many many universes overlapping each other in multiple levels of superposition. in this way you develop a complex causality that is fully deterministic, while still enabling time travel, just not back in the actor's own lightcone.
This makes "Time travel" more a kind of multiverse travel, with a "fuzzy" probability that you might or might not arrive in the desired universe, with absolute precision being impossible, due to physics induced uncertainty. For the vast majority of cases, this will not present a problem. For instance, if you and a quantum duplicate you in a different universe exchange places as a result of FTL travel over your own lightcone, the vast majority of possible differences will be very similar, or not important. EG, the photons emitted from the lightbulbs might be different, but the macro-events are the same. That is a sufficiently large enough difference to be a distinct quantum universe, but if you exchange places with that other you, neither of you will notice, nor care.
Your target location in time and space is going to be a fuzzy probability of potential universes meeting those criteria, and your arrival or lack of arrival is just going to be another probability function. In universes where "somebody" arrives, there's a specific probability that it will be you-- etc.
This leads to some interesting plot features that could develop. Naturally, the more macro-changing the reason for the FTL travel (EG, are you going to pick up and deliver cargo-- or are you going to avert or cause a war?) the more affected the probability that a ship from a parallel universe is what will arrive at the destination-- and the more risky it is to engage in the FTL travel (If you are concerned about the politics of who arrives) This may lead to laws saying FTL is only for mundane trips, and not politically sensitive ones, to avoid having Bizzarro Superman arrive on the scene. This self-feeds, because the most probable intersections for travel will be with universes that consider political FTL to be acceptable-- which are the most likely to have sent a shep, and be expecting one to arrive. The deal is, you wont know WHO arrives, until they do. When going to negotiate an essential peace treaty, the LAST thing you want is "Evil Kirk" showing up to do the negotiations. To prevent that, political envoys are forbidden from using FTL travel as a precaution-- etc. (this is amusing, because the last person that the Evil Universe wants to show up at the decisive space battle is pacifist spacehippy kirk. When both sets of universes obey the law, the probabilities of these exchanges happens is diminished considerably.)
The best that the flight computer can do is attempt to navigate to a parallel universe where the known quanta are as close to a match to the universe of origin as is possible. As I pointed out, most universes that intersect via FTL this way will have very small divergent features, and the people exchanging universes will not have anything to care about, other than on a philosophical level. (Is that really still my husband? etc.) It is when there are huge potential ramifications to the interchange that things get dicey, and people obeying the kinds of civil laws I pointed out above would help alleviate that problem. considerably. The universes that obey the law will intersect with those that dont, only on very rare occasions.
This sidesteps many issues. For example, the cargo leaving one planet, headed for another, arrives at a universe different from its origin-- but a virtually identical cargo vessel arrives at the desired planet, and makes the delivery. To the people shipping the material, and the people recieving the material, it does not matter the subtle differences, and the reality that they are not, in actuallity, the "same" container. Do you care about the exact quantum states of the atoms in your ham sandwich before you eat it-- or are you more concerned it about it actually being ham, and not some mystery meat? That's the takeaway there. You dont NEED the "same" container. You just need a container that has what you want inside-- FTL shipping and receiving of products and raw materials would still be very useful economically, even with this caveat. The same would be true for sending people. Who cares if you get the Doctor Hawking from this universe, or the "basically identical universe except for the difference of spin state of a handful of photons someplace" version? Both have the same life experiences on the macro scale, both have the same intellect, and will behave in excactly the same ways. They are virtually identical. The subtle reality that they are not, in actuality, the "Same Person" in the causal sense, allows FTL and timetravel to reconcile.
Some universes may discover that FTL travel in this fashion is possible-- and choose not to allow it at all. Those universes will not intersect with universes that do engage in it, except on very rare occasions. Others may decide that the practical benefits of FTL travel far outweigh the philosophical consequences, and engage in it vigorously. These universes will exchange matter and information via FTL incursions very regularly, and will likely be quite homogenous as a result.The vast majority of FTL solutions will fall in this space for these universes. Then you have those that allow FTL for politically charged transport-- and those are where the interesting story can happen.
Once you eliminate the mental need for the universe to be a singular thing, and allow multiple possible origins and destinations, with fuzzy probabilities about which ends up tied to which, you obviate nearly all of the paradoxes involved, and still have "Functional" FTL travel.