There are no 5000-7000ft. mountains this time of year not to have an ice axe with you. You could hike Artist Point at Mt. Baker which is 5000 but with all the new snow this week I would take snowshoes.
Above 5000 feet there is 20-50 inches of new fresh unconsolidated snow. Right now I am serious thinking of cancelling my Hood-Old Chute (I am leading a group of 9 experienced mountaineers) trip for this Saturday due to high avy danger.
Without an ice axe or snowshoes you are going to have a hard time above 5000' especially if you are going solo. Pilchuck maybe, but even that is very questionable. Be patient and wait a week or two for snow to consolidate. Personally I would advise either Mount Zion, Mount Walker, Mount Si, Rattlesnake Ridge, Temple Mountain, the Chuckanuts or East Tiger. With that left over time stop off at REI or another gear store and maybe find a couple snowshoes on sale.
As for the axe, purchase one and then try to either take a class or link up with someone who is willing to train you on how to self-arrest. With the ski areas being closed this is the perfect time for that. Good luck on your journeys.