I have had gout and kidney stones (both caused by the same thing).
Drink lots of water. This is particularly true after eating. This is the most important advice.
The cherry advice is good. You don’t need many, maybe five a day. If cherries are out of season, try cherry juice…again not much is needed and you should probably even water it down a lot.
Lose weight. This will help both the gout and the blood pressure. Maybe you should do this to some degree before continuing your weight training. Just monitor things and see what happens.
Avoid foods that are very high in phospherous. One major culprit is soft drinks, particularly colas. You should eliminate these from your diets.
Also, eliminate alcohol for now, until your weight is down. Beer is particularly bad. Later it might be okay to experiment a bit.
I now take whey powder daily, 50 to 60 grams of protein from that source and have no problem.
It would probably be good to cut back on high impact cardio until you lose some weight. There are plenty of other options. Biking is good. So is the rowing machine if your back is strong.
Have a blood test done in a few months and if all is back to normal (and I suspect if you work hard, it will be), you can try running again.
The weight training per se will not effect your gout, the added protein may. If your not using whey powder, try it and see what happens.
Gout attacks are triggered by a variety of conditions and people seem to have different triggers. In summary, drink water and then more water; avoid soft drinks that contain phospherous, control your weight, avoid alcohol, eat healthy!
Hope it all works out for you.