I can't add much to Valkine's excellent suggestions, but if you are looking for sources beyond the First Crusade, there are also a couple of recent books by Michael Fulton, Artillery in the Era of the Crusades: Siege Warfare and the Development of Trebuchet Technology (Brill, 2018) and Siege Warfare during the Crusades (Pen and Sword Military, 2019).

Nicholas Morton also recently published The Crusader States and Their Neighbours: A Military History, 1099-1187 (Oxford University Press, 2020).

The two classic books about crusader warfare are maybe a bit old now, but they are: R.C. Smail, Crusading Warfare, 1097-1193 (Cambridge University Press, 1956, 2nd ed., 1995), and Christopher Marshall, Warfare in the Latin East, 1192-1291 (Cambridge University Press, 1992).

For questions of supply and organization, see John H. Pryor, ed., Logistics of Warfare in the Age of the Crusades (Ashgate, 2006), especially the chapter by Bernard S. Bachrach, "Crusader logistics: from victory at Nicaea to resupply at Dorylaion".

John France is usually my go to for warfare in this period, but I can see you've already mentioned him so here are a few people I would recommend reading as well:

You should definitely add Steve Tibble to your list of scholars to check out, he's one of the most interesting people currently working on the military history of the crusades. I read his book The Crusader Armies: 1099-1187 earlier this year and found it very thought provoking. I don't 100% agree with everything he says, but I admire his willingness to push out new theories and ideas. I haven't read his book The Crusader Strategy yet but I'm very much looking forward to it.

Besides Tibble, you might find some interesting bits of information in Christopher Tyerman's How to Plan a Crusade. Despite it's more pop-history title this is a pretty dense work that I would recommend using as more of a reference book than reading cover to cover (mostly because it's a bit boring as a cover to cover read, not because it's particularly impenetrable). This book covers a wide geographic and chronological span, but you'll probably find something of interest in it.

The First Crusade has one of the more straightforward sets of primary sources to be familiar with. Military historians tend to prefer the Gesta Francorum, but I've always been a bit of a sucker for Raymond of Aguilers as an underappreciated source. Fulcher of Chartres is of course a major eyewitness account. Back in Europe you have Albert of Aachen, who is generally held to be mostly trustworthy, plus a whole slew of others who are held in varying levels of esteem. On the much more obscure end, there's always Ralph of Caen's Gesta Tancredi, who almost nobody outside of niche Crusades studies ever seems to talk about.

If you really want to get in the weeds I'd recommend Colm Flynn's PhD thesis Crusader artillery, 1097-1148 : typology, terminology and character, which is freely available online.

Edit: I forgot to include relevant works on siege warfare. Peter Purton's History of the Early Medieval Siege c.450-1220, is probably the stand out work on the subject right now, but R. Rogers' Latin Siege Warfare in the Twelfth Century is also excellent a bit more focused around this time period.