I am an Airbus FBW fan. I have around 8,500TT on A320, A321 and A330 and a year on B737.
Owing to a very long established and respected Airbus airline suddenly going out of business because of mismanagement; I had to take a job with B737 classics after about 15 years on the Airbus.
I could fly the Boeing because I had flown other conventional basic jets and turbo-props in previous airlines. So pitch trimming all the time and holding back pressure in turns etc. was automatic to me and I had no problem flying the Boeing.
However, I found the B737 to be a real step backwards from the Airbus in terms of its lack of sophistication and basic nature. The 737 cockpit is very cramped, and the automatics logic is very strange after the Airbus FBW; which is an order of magnitude better. The Boeing auto-thrust logic is also odd, and needs to be disconnected if flying manually - a far cry from the excellent Airbus.
Other Boeings seem better but I have only briefly flown B757/767/777 in the SIM. All Boeings still have a bloody great yoke in the way of the instruments though......
B737 have a "Captain plus secretary" attitude and layout, and the F/O (secretary) has a lot to do on the Classic overhead panel, whereas the Airbus is laid out for two full pilots, with all controls available to both sides. Airbus have a much more logical overhead panel layout as well.
If I had a free choice, I would go for Airbus FBW !
But you probably won't have a free choice. I am sure that you will be able to fly a Boeing - I could after many years on the Airbus - but they might well put you on the Airbus fleet to avoid having to put you on a complete new type rating ?
Good luck !