We can use present simple to talk about a planned or scheduled event in the future.
My flight is at 10:45am tomorrow
I am leaving tomorrow
I am meeting ... today
This could be simple present meaning it's happening now and going on all day, or it could be a planned event that will happen today- to be planned, it must be later than now.
Inserting after ages into the middle of this sentence does not really work. It would be better to say:
I am meeting my ex-girlfriend today: I haven't seen her for ages.
Moving on to the the past simple/past continuous question:
I was meeting my ex girlfriend and the conversation felt awkward.
This is talking about an event in the past. It's over now, so you should use present simple.
I met my ex girlfriend and the conversation felt awkward.
You would normally only use past continuous if you were in the middle of one thing when something else happened, or it was planned but was cancelled. For met/was meeting:
I was meeting my ex-girlfriend when my new girlfriend walked in.
I was meeting my ex-girlfriend but she got held up at work.
And for felt/was feeling...
I met my ex-girlfriend and the conversation was feeling awkward until we'd had a couple of drinks.