The thing with incomplete wars, and this case in particular, is that you have gained a few cities, but these cities are swamped in foreign culture that will take a lot of effort to push back.
Say you take coventry and nottingham, this would remove cultural pressure on london and hastings. This would also free up a part of your garrisons to join your main army. A city in the backlines only needs one unit.
To decide whether to continue the war, what units and how many of them do you have? what do they have? did you kill alex's army?
This still seems to be longbows against rifles (soon infantry!). If that is the case, and you are lacking units, then you did not build enough. General advice on this: do not stop building units too early. Granted that is very vague, and difficult to judge. For starters you might have difficulties to say if you have enough troops to finish your wars. That is perfectly normal - but your reaction should then be to continue building units until you are reasonably certain you have enough.
Trying to do many things at once (e.g. war, building infrastructure, teching ...) usually means you divert attention from one of them. Especially for wars this is dangerous, as that will draw your war out, increasing the negative effects of a war (unit cost, danger from enemy units, war weariness) without getting you what you want: more land.
regarding your economy, it is doing too well. You are in a war, possibly have too few units, and can sustain 50% research? You have the econ tech you need (communism), so recovery will just be to let your cities grow. Your new cities could arguably be building units, instead of infrastructure. Your core cities should be whipping, especially those that are unhappy.
One other thing: some city (preferably one with good production) should build a workboat for portland.
One thing that you must do is to look for resource trades from the other continents, especially for happiness resources. You need them badly.
War is hard lol i feel like I'm really slow with my conquering, and it's hard to protect the cities I take.
Optimally you take out the enemy army first, to reduce the chance for counter attacks. Only here that will be of mediocre usefulness as alex is a mad unit spammer who will even attack at relatively low odds, thus posing a greater threat. Also it is better to take cities quickly. With the kind of advantage you have, attacking with only cavs would probably have been better (in this case they are maybe slightly better than cuirs, but cuirs would have worked really well too.), as they are much faster. Granted, you can not bombard, but a cav (or cuir) attacking a city with maybe 60% cultural defence only defended by longbows and pikes gets good odds already