I think you would be ok to apply for cards. You just need to be a little bit more patient. How about one card every 3 months? And start with cards you think you'll keep (either no fee or low fee and worth the cost) that way over time you can let the earlier cards age and show issuers you aren't just a churner. For example I'd start with SPG Amex with only $65 fee, you can keep that one for years. There are rumors the bonus on that card will be going up 30k pretty soon. Then in a few months if the Freedom card bonus comes back up take that one. Not huge bonuses, but cards you can keep. Then in a few more months move on to some other cards.

Also, how much card apps impact mortgage apps is somewhat of an open question. There are a few threads here discussing this.